IC-NRLF 


33    23D 


ELECTRICAL 
ENGINEERING  PROBLEMS 

PART  I 

DIRECT   CURRENT   CIRCUITS 
AND  APPARATUS 

PART  II 

ALTERNATING  CURRENT  CIRCUITS 
AND  APPARATUS 


BY 

F.  C.  CALDWELL,  A.B.,  M.E. 

Professor  of  Electrical  Engineering  at  the  Ohio  State  University, 
Fellow  American  Institute  of  Electrical  Engineers 


FIRST  EDITION 


McGRAW-HILL  BOOK  COMPANY,   INC. 
239  WEST  39TH  STREET,  NEW  YORK 

6  BOUVERIE  STREET,  LONDON,  E.  C. 
1914 


COPYRIGHT,  1914,  BY  THE 
McGRAW-HILL  BOOK  COMPANY,  INC. 


Stanbope  jpress 

F.    H.  GILSON  COMPANY 
BOSTON,  U.S.A. 


PREFACE 

This  little  book  of  problems  is  offered  as  a  system  of  exercises, 
suitable  for  use  with  any  of  the  available  textbooks  or  with  a 
lecture  course.  Most  of  the  problems  have  been  used  in  the  classes 
of  the  author.  A  novel  feature  is  the  stating  of  the  time  needed 
to  perform  the  actual  solving  of  each  problem.  This  should  prove 
useful  not  only  to  the  instructor  in  the  assignment  of  problems, 
but  also  to  the  student  as  a  measure  of  his  own  speed.  In  most 
of  the  problems  the  time  given  is  that  which  was  actually  used  by 
an  undergraduate  student.  It  will,  however,  probably  be  found 
necessary  to  allow  more  time  to  the  average  class  than  is  here 
indicated. 

Sheets  giving  the  answers  to  the  problems  will  be  provided  in 
such  quantities  as  may  be  needed,  but  only  to  instructors. 

F.  C.  C. 

COLUMBUS,  OHIO. 
January,  1914. 


285770 


CONTENTS 


PART  I 
DIRECT-CURRENT  CIRCUITS  AND  APPARATUS 

CHAPTER  PAGE 

I.   E.M.F.,  CURRENT,  CONDUCTANCE  AND  RESISTANCE 3 

II.   WIRES,  WIRE  TABLES,  RESISTIVITY  AND  TEMPERATURE  COEFFI- 
CIENT   . 6 

III.  POWER  AND  WORK 8 

IV.  MEASUREMENT  OF  CURRENT  AND  E.M.F 10 

V.   MAGNETISM  AND  MAGNETIC  CIRCUITS 11 

VI.   MAGNET  WINDINGS  AND  MAGNETS 15 

VII.   GENERATION  OF  ELECTRO-MOTIVE-FORCE,  ARMATURE  DROP.  .  18 

VIII.   ARMATURE  WINDINGS 21 

IX.   ARMATURE  CIRCUIT  CALCULATIONS 23 

X.   ARMATURE  REACTIONS 25 

XI.   MAGNETIZATION  CURVES 28 

XII.   CHARACTERISTICS 30 

XIII.  HEATING  AND  RATED  CAPACITY  OF  DYNAMOS 33 

XIV.  DYNAMO  LOSSES  AND  EFFICIENCIES 35 

XV.   MOTORS 39 

PART  II 
ALTERNATING-CURRENT  CIRCUITS  AND  APPARATUS 

I.   INDUCTANCE  AND  INDUCED  E.M.F 47 

II.   QUANTITY  AND  CAPACITY,  CONDENSERS 51 

III.  ALTERNATORS  AND  WAVE  FORMS 53 

IV.  ALTERNATING  CURRENT  IN  INDUCTIVE  CIRCUITS 55 

V.   INDUCTIVE  CIRCUITS  IN  SERIES  AND  PARALLEL 59 

VI.   CAPACITY  AND  INDUCTIVE  CIRCUITS,  RESONANCE 66 

VII.  SINGLE-PHASE  POWER,  WATTMETERS 70 

VIII.   POLYPHASE  SYSTEMS  AND  POWER 73 

IX.   TRANSFORMERS,  GENERAL 78 

X.   TRANSFORMER  DIAGRAMS  AND  REGULATION 84 

XI.   SYMBOLIC  EXPRESSIONS  AND  METHODS 88 

XII.   ALTERNATOR  REACTIONS  AND  REGULATIONS 91 

XIII.  SYNCHRONOUS  MOTORS  AND  GENERATORS 93 

XIV.  SYNCHRONOUS  CONVERTERS 97 

XV.   POLYPHASE  INDUCTION  MOTORS 100 

TABLES 103 

ABBREVIATIONS 

B,  gausses.  C,  conductors,  c.p.,  candle-power,  e.m.f.,  electro-motive- 
force.  H,  magneto-motive-force  per  centimeter  length,  h.p.,  horse-power, 
i,  current,  kw.,  kilowatt,  kv-a.,  kilovolt-ampere.  1,  length.  N,  turns. 
R,  resistance,  r.p.m.,  revolutions  per  minute. 

As  subscripts,     a,    armature;   g,  gap;   m,  magnet;   s,  shunt,  f,  series  field. 


PART  I 
DIRECT  CURRENT  CIRCUITS  AND  APPARATUS 


ELECTRICAL  ENGINEERING  PROBLEMS 


PART   I 
CHAPTER  I 

ELECTROMOTIVE   FORCE,    CURRENT,    CONDUCTANCE 
AND    RESISTANCE 

1.  Given  three   110-volt  tungsten  lamps  with  resistances  of 
50,  120  and  200  ohms  in  series,  with  330  volts  applied  across  the 
outside,  what  will  be  the  difference  of  potential  around  each  lamp? 
(3  raw.) 

2.  Given  circuits  of  4  and  6  ohms  in  parallel,  and  in  series  with 
these  a  circuit  of  7.6  ohms.     What  current  will  be  sent  through 
this  combination  of  circuits  by  120  volts?     If  each  of  the  resistances 
be  halved,  what  will  the  current  be?     (2  ram.) 

3.  Given  two  circuits  of  5  and  7  ohms  in  parallel  between 
two  points  A  and  B,  and  in  series  with  these,  10  ohms  between 
B  and  another  point  C.     If  310  volts  be  applied  between  A  and 
C,  required  the  volts  between  A  and  B,  and  between  B  and  C 
and  also  the  current  flowing.     (4  ram.) 

4.  Five    625-ohm,    20-candle-power    and    four    392-ohm,   32- 
candle-power,  125-volt  tungsten  lamps  are  all  in  parallel.     The 
resistance  of  the  circuit  connecting  them  to  the  generator  is  2 
ohms.     Find  the  conductance  of  each  lamp  and  of  the  whole 
group,  the  current  taken  by  each  lamp  and  the  voltage  required 
at  the  generator  to  give  the  lamps  their  proper  current.     (8  ram.) 

5.  Given  three   circuits  of  5,   3   and  1T3T   ohms  respectively 
in  series,  and  in  series  with  these,  two  parallel  circuits  of  5  and 
6  ohms.     With  360  volts  applied  to  the  outside  terminals,  what 
will  be  the  pressure  on  the  3-ohm  circuit  and  what  current  will 
flow?     If  this  circuit  be  changed  from  3  to  15  ohms,  in  what  ratio 
will  the  drop  around  the  5-ohm  circuit  be  changed?     (3  ram.) 

6.  The  voltage  between   the   terminals   of  a   6-foot  piece  of 
wire  is  50  and  its  resistance  is  100;  for  the  calibration  of  a  volt- 

3 


4  V. :  I-*":  ttttCftl&f     ENGINEERING  PROBLEMS 

meter  12 J  volts  are  wanted;  how  many  ohms  of  the  resistance 
of  the  wire  and  what  length  must  be  included  between  the  ter- 
minals of  the  voltmeter?  (The  voltmeter  is  supposed  to  have  an 
infinite  resistance.)  (1  min.) 

7.  Two  resistances  of  50  and  100  ohms  are  connected  in  parallel 
between  two  points  A  and  B,  between  which  the  e.m.f.  is  100  volts; 
what  current  will  flow  in  each  circuit?     If  the  100-ohm  circuit 
be  reduced  to  0.05  ohm,  what  current  will  flow  in  each?     (2  min.} 

8.  Circuits  P,  Q  and  R  are  in  parallel,  as  are  also  circuits  S 
and  T;   these  two  groups  are  in  series.     If  270  volts  are  applied 
to  the  outside  terminals  of  the  combination,  find  the  conductance 
and  resistance  of  each  group  and  the  current  in  each  circuit,  the 
resistances  of  the  circuits  being  3,  5,  7J,  20  and  30  ohms  respec- 
tively.    (7  min.) 

9.  A  resistance  of  42  ohms  between  A  and  H  is  tapped  at 
six  equidistant  points,  B,  C,  D,  E,  F  and  G;   four  switches  are 
placed  between  A,  C,  E  and  G,  and  one  bus  bar,  and  four  between 
B,  D,  F  and  H,  and  the  other  bus  bar.     Find  the  conductance  and 
resistance  between  the  bus  bars  with  each  of  the  following  six 
groups  of  switches  closed  —  A,  B  and  G ;  A,  C  and  H ;  A,  D  and 
G;  A,  B,  C  and  H;  A,  B,  C,  G  and  H;  all  closed.     (8  min.) 

10.  Four  points  A,  B,  C  and  D  are  at  the  successive  corners 
of  a  square  and  resistances  are  connected  between  them  as  fol- 
lows: AB,  2  ohms;  BC,  6  ohms;  CD,  12  ohms;  BD,  100  ohms. 
What  resistance  must  be  placed  between  D  and  A  in  order  that 
no  current  shall  flow  through  the  circuit  BD  when  10  volts  is 
applied  between  A  and  C?     What  if  100  volts  be  applied?     (2  min.) 

11.  The  field  of  a  15-kw.  shunt  motor  has  a  resistance  of  25 
ohms;   what  current  will  flow  when  it  is  connected  in  on  a  con- 
stant-potential 125-volt  circuit?     If  the  armature,  the  resistance 
of  which  is  0.04  ohm,  was  connected  in  parallel  (not  running), 
what  current  would  flow  through  it?     What  would  then  be  the 
current  through  the  field?     What  would  happen  to  the  armature? 
If  a  5-ohm  rheostat  were  included  in  the  armature  circuit,  what 
would  be  the  total  current  taken  by  the  motor?     (4  min.) 

12.  A  70-cell  storage  battery,  designed  to  give  100  amperes 
for  8  hours,  has  a  resistance  of  0.0005  ohm  per  cell.     When  the 
charging  of  the  battery  is  nearing  completion,  the  cells  have  an 
electromotive  force  of  2.5  volts  each.     If  a  220- volt  generator  is 
being  used  for  charging  at  100  amperes,  what  voltage  must  be 


ELECTROMOTIVE  FORCE  5 

used  up  in  resistance,  and  how  many  ohms  will  be  needed?     (3 
min.) 

13.  The  shunt-field  resistance  of  a  10-kw.,  120-volt  generator 
is  to  be  measured.     A  rheostat  is  in  series  with  it  on  the  120-volt 
circuit.     The  current  is  2  amperes  and  the  drop  around  the  field 
is  106  volts.     What  is  the  resistance  of  the  field,  of  the  rheostat 
and  of  the  combination?     (2  min.} 

14.  In  measuring  a  certain  railway  current  a  "shunt"  having 
a  resistance  of  0.00025  ohms  is  connected  in  the  circuit,  and  a 
millivoltmeter  attached  to  its  terminals  reads  0.05  volts.     What 
is  the  value  of  the  current?    Why  should  the  resistance  in  this 
shunt  be  so  low?     (2  min.) 


CHAPTER  II 

WIRES,   WIRE   TABLES,    RESISTIVITY   AND    TEMPERATURE 
COEFFICIENT 

Note.  —  Unless  otherwise  stated  the  resistivity  of  copper  at 
atmospheric  temperature  may  be  taken  as  11  (based  on  the 
circular  mil-foot). 

Note  also  that  in  the  American,  or  Brown  &  Sharp  wire  gauge 
the  diameters  of  the  wires  from  No.  6  to  No.  12  are  approximately 
the  reciprocals  of  the  numbers  expressed  in  inches;  thus  No.  10  is 
TV  inch  or  100  mils  diameter  (actual  102).  Also  that  the  diameter 
doubles  for  every  6  numbers  and  the  area  for  every  3  numbers, 
and  the  area  increases  10  times  for  10  numbers.  Use  the  B.  &  S. 
wire  table  on  page  103. 

1.  By  referring  to  the  diameters  of  the  sizes  from  6  to  12,  ob- 
tained without  consulting  the  tables,  determine  the  approximate 
diameter  in  mils  and  the  area  in  circular  mils  of  the  following  wires : 
Nos.  35,  23,  5,  1.     Determine  also  the  per  cent  errors  in  area  that 
would  be  made  in  using  these  approximations.     (See  wire  table 
for  exact  sizes.)     (15  min.) 

2.  The  area  of  a  No.  10  wire  being  10,400,  what  will  be  the 
approximate  area  of  a  No.  4  wire?     Of  a  No.  13?     Of  a  No.  20? 
(2  min.) 

3.  If  one  dimension  of  a  rectangular  wire  is  to  be  twice  the 
other,  what  must  they  be  in  inches  to  replace  a  No.  6  wire?     What 
will  be  the  area  in  sq.  mm.?     (4  min.) 

4.  How  many  square  mils  and  how  many  circular  mils  in  a 
wire  \  inch  by  fV  inch?     Also  if  this  wire  is  rounded  at  the  cor- 
ners with  a  radius  of  20  mils,  what  will  be  its  area  in  circular 
mils?     (4  min.) 

5.  Without  consulting  the  tables  determine  what  B.  &  S.  wires 
will  have  to  be  used  in  circuits  requiring  the  following  areas  of 
copper:  40,000,  3600  and  100  circular  mils.     Also  for  the  following 
diameters  in  mils:  7.5,  19,  62,  135  'and  240.     (6  min.) 

6.  Given  a  copper  wire  300  feet  long  and  6529  circular  mils 
cross-section,  No.  12  B.  &  S.,  find  the  volts  to  give  25  amperes. 
Also  if  the  length  be  made  600  feet.     (2  min.) 

6 


WIRES,  WIRE  TABLES,  ETC.  %     7 

7.  2000  feet  of  No.  10  wire,  102  mils  in  diameter,  has  a  resist- 
ance of  2  ohms  at  20°  C.     What  is  the  resistivity  of  the  material? 
What  might  the  material  be?     (1  min.) 

8.  3500  feet  of  No.  25  wire  of  a  certain  material  used  for  con- 
ductors has  a  resistance  at  20°  C.  of  175  ohms.     Its  area  is  320 
circular  mils.     What  is  the  resistivity  and  what  is  the  material? 
(2  min.) 

9.  Given  the  temperature  coefficient  of  copper  as  0.004,  and 
the  resistance  of  a  circular  mil-foot  at  25  degrees  as  10.55;  required 
the  resistance  of  a  No.  18  wire  150  feet  long  at  55°  C.     (2  min.) 

10.  At  the  working  temperature  of  70°  C.,  the  field  of  a  shunt 
dynamo  has  a  resistance  of  100  ohms;   how  many  feet  of  No.  14 

B.  &  S.  German  silver  wire,  whose  resistivity  is  290,  must  be  inserted 
in  series  to  keep  the  field  current  the  same,  when  the  machine  is 
started  at  a  room  temperature  of  10°  C.?     (5  min.) 

11.  Required  the  e.m.f.  necessary  to  send  25  amperes  through 
one  mile  of  No.  10  copper  wire.     If  a  generator  supplied  500  volts 
at  one  end  of  this  circuit,  what  e.m.f.  would  be  available  to  run 
a  motor  at  the  other?     Would  this  be  an  economical  transmission? 
(4  min.) 

12.  A  lighting  plant  is,  during  the  day,  supplying  two  250- 
ohm  incandescent  lamps  connected  in  series,  at  a  distance  of  500 
feet  from  the  station,  with  current  at  250  volts,  the  conductors 
being  one  inch  in  diameter.      At  evening,  500  additional  pairs  of 
two  lamps  (in  series)  are  placed  in  parallel  with  the  pair  already 
burning.     Required  the  current  flowing  through  the  two  lamps 
during  the  day,  the  total  amount  of  the  night  load  in  amperes  and 
the  change  in  the  current  through  the  two  original  lamps;   what 
would  this  change  be  if  the  resistance  of  the  feeding  wires  were 
zero?     (6  min.) 

13.  A  voltmeter  has  a  resistance  with  the  leads  of  one  ohm, 
and  is  to  be  used  at  a  distance  of  25  feet,  and  connected  with  a 
copper  wire  having  a  temperature  coefficient  of  0.004  per  degree 

C.  Required  the  area  and  size  of  wire  necessary  in  order  that  an 
increase  of  8  degrees  from  24°  C.  may  not  cause  an  error  of  over 
\  per  cent.     Take  resistivity  for  24  degrees  as  10.5.     Note  that 
the  voltmeter  is  calibrated  with  the  leads  in  series,  and  also  that 
the  readings  are  independent  of  the  temperature  of  the  instru- 
ment.    (5  min.) 


CHAPTER  III 
POWER   AND    WORK 

1.  50    amperes    at   110  volts   give   how   many   horse-power? 
Required  the  current  at  500  volts  to  give  the  same  horse-power; 
also  at  1000  volts.     (2  min.) 

2.  At  8  cents  per  kilowatt-hour  how  much  will  it  cost,  per  week 
of  60  hours,  to  run  a  motor  having  an  average  load  of  4  horse- 
power and  an  average  efficiency  of  80%?     (3  min.) 

3.  A  20-can die-power  tungsten  incandescent  lamp  takes  1.3 
watts  per  candle-power;    a  common  price  for  this  purpose  is  10 
cents  per  kw.-hour;  at  this  price,  what  will  it  cost  to  run  8  lamps 
for  three  hours?     (2  min.) 

4.  What  must  be  the  horse-power  delivered  by  an  engine  to 
run  by  belt  a  generator  feeding  500  J-ampere  lamps  at  110  volts? 
Four  volts  are  lost  in  the  line,  the  efficiency  of  the  generator  is 
90%  and  the  loss  in  the  belt  is  1%.     (4  min.) 

5.  A  motor  at  1000  feet  from  the  generator  requires  20  amperes 
at  500  volts.     The  wire  used  is  a  No.  6.  .  Required  the  e.m.f .  at 
the  generator  and  the  per  cent  of  the  volts  lost;   also  the  power 
lost  and  the  per  cent  of  the  power  lost.     (5  min.) 

6.  Same  as  problem  5  but  using  No.  4  wire.     (5  min.) 

7.  What  will  be  the  loss  in  pressure  and  in  watts  in  trans- 
mitting 100  horse-power  at  500  volts  through  a  No.  0000  wire, 
taking  the  resistance  for  30°  C.  as  0.05086  ohms  per  1000  feet,  and 
the  distance  being  one  mile?     What  are  the  respective  per  cents  of 
e.m.f.  and  power  lost?     (5  min.) 

8.  A  220-volt  motor  with  80%  efficiency  gives  8.94  horse- 
power, is  1000  feet  distant  from  the  generator  and  is  wired  with 
a  No.  6  wire;   how  many  horse-power  are  lost  in  the  circuit,  and 
at  what  voltage  must  the  generator  run  in  order  that  the  motor 
may  have  its  proper  pressure?     What  per  cent  of  the  delivered 
e.m.f.  and  power  is  lost  in  the  transmission?     (6  min.) 

9.  A  test  with  a  Prony  brake  shows  that  a  certain  motor  is 
giving  132,000  foot-pounds  per  minute.     The  input  is  7.46  amperes 
at  500  volts.     What  is  the  efficiency?     (2  min.) 

8 


POWER  AND  WORK 


9 


10.   For  the  direct  driving  of  a  factory  the  following  list  of 
motors  is  necessary: 


Number 

H.  P. 

Efficiency 

1 

100 

92% 

2 

30 

90 

6 

20 

88 

10 

5 

83 

20 

2 

75 

On  this  circuit  the  full-load  line  loss  is  5%  of  the  power  delivered 
to  the  motors. 

There  are  also  100  32-candle-power  tungsten  incandescent 
lamps  taking  1.25  watts  per  candle  power,  and  10  flaming  arcs 
taking  550  watts  each.  Line  loss  is  neglected  on  the  lighting 
circuit. 

What  will  be  the  kilowatt  capacity  of  the  direct-connected 
generator  and  what  horse-power  must  the  engine  give  if  the 
dynamo  efficiency  is  94%?  (11  min.) 


CHAPTER  IV 
MEASUREMENT    OF   CURRENT   AND    E.M.F. 

1.  It  is  required  to  measure  5542  amperes  by  means  of  a  resist- 
ance of  5  X  10~6  ohms  and  a  voltmeter;   draw  a  diagram  of  the 
connections,  and  show  what  would  be  the  reading  of  the  voltmeter 
and  what  power  would  be  lost  in  the  shunt.     Why  not  use  a  0.001- 
ohm  resistance?     (5  min.} 

2.  Five  lamps  having  resistances  of  193,  203,  207,  197  and  200 
are  put  in  series  across  a  railway  circuit,  and  a  150- volt  voltmeter 
of  infinite  resistance  is  connected  around  the  200-ohm  lamp;  it 
reads  97  volts;  what  is  the  pressure  on  the  circuit?     What  would 
be  the  reading  for  this  pressure  if  the  resistance  of  the  voltmeter 
were  only  500?     How  low  a  resistance  in  the  voltmeter  could  be 
left  out  of  the  calculation  for  an  accuracy  of  \  per  cent?     (17  min.} 

3.  It  is  desired  to  use  a  50-millivolt  meter  as  a  200-ampere 
meter  on  a  500-volt  circuit.     What  resistance  must  the  shunt 
have?     Same  for  a  0.2-ampere  meter.     What  will  be  the  voltage 
on  the  instrument  if  the  shunt  is  opened  between  the  millivolt- 
meter  terminals?     (3  min.} 

4.  How  low  could  the  resistance  of  the  millivoltmeter  used  in 
each  case  of  problem  3  be,  without  introducing  an  error  of  more 
than  TV  per  cent?     (6  min.} 

5.  A  150-volt  meter  having  a  resistance  of  16,000  ohms  is  to 
be  used  with  a  multiplier  as  a  600- volt  instrument.     What  resist- 
ance must  the  multiplier  have?     (3  min.} 

6.  If  the  readings  on  a  150-volt  meter  can  be  estimated  accu- 
rately to  0.2  of  a  division,  what  per  cent  of  accuracy  can  be  obtained 
at  1.5,  15  and  150  volts  respectively?     (2  min.} 


10 


CHAPTER  V 
MAGNETISM   AND    MAGNETIC    CIRCUITS 

1.  Upon  cross-section  or  coordinate  paper  construct  the  mag- 
netization curves  from  page  105,  for  cast  iron,  cast  steel,  wrought 
iron  and  sheet  steel.     Plot  with  magnetizing  force  H  for  abscissae, 
and  density  B  for  ordinates.     Use  the  same  scale  for  all  the  curves 
and,  while  taking  one  that  is  easily  read,  choose  it  so  as  to  occupy 
with  the  curves  as  much  of  the  sheet  as  practicable.     Carry  the 
curves  to  about  H  =  200.     (30  min.) 

2.  Given  a  ring  of  cast  iron,  50  cm.  in  circumference;  required 
the  gilberts  necessary  to  give  6500  gausses,  and  also  the  turns, 
if  the  current  is  5  amperes.     (2  min.) 

3.  If  a  cut  be  made  in  the  ring  of  the  last  problem  and  spread 
out  to  J  cm.  gap,  what  will  be  the  total  number  of  turns  necessary 
for  the  same  density?     (2  min.) 

4.  A  forged  ring  has  a  mean  diameter  of  45  centimeters,  and  an 
area  of  30  sq.  cm.     The  flux  is  450  kilomaxwells.     Required  the 
gilberts  and  the  ampere  turns  necessary.     (5  min.) 

5.  Required  the  reluctance  in  oersteds  of  the  above  ring  at  the 
density  used;  also  the  permeability.     (4  min.) 

6.  Required  the  reluctance  in  oersteds  of  a  cast-iron  ring  of  the 
same  dimensions  as  in  problem  5  and  having  a  permeability  of 
200.     What  density  does  this  indicate?     Compare  with  the  reluct- 
ance found  in  problem  5.     (8  min.) 

7.  A  horseshoe-shaped  magnet  for  hoisting  rails  is  forged  of 
wrought  iron  and  has  a  magnetic  path  40  cm.  long;  the  oxide 
on  the  rail  is  0.1  mm.  thick  (note  the  two  gaps),  and  the  path 
through  the  rail  is  15  cm.     The  density  in  the  magnet  is  16,000 
gausses,  and  the  cross-section  of  the  rail  is  double  that  of  the  mag- 
net.    Magnetizing  force  H  for  both  irons  is  to  be  taken  as  for 
wrought  iron.     Required  the  number  of  turns  necessary  in  the 
winding  if  the  exciting  current  is  two  amperes.     (4  min.) 

8.  A  cast-steel  magnetic  clutch  for  driving  a  pulley  must  pull 
its  armature  up  when  it  is  1  mm.  away  (note  the  two  gaps).     Its 

11 


12       ELECTRICAL  ENGINEERING  PROBLEMS 

magnetic  path  including  the  armature  is  30  cm.  To  do  this,  the 
density  through  the  air  must  be  8000  gausses,  which  will  require 
11,000  in  the  magnet  and  armature.  How  many  turns  will  be 
necessary  with  an  exciting  current  of  3  amperes?  What  would 
the  density  become  when  the  armature  was  pulled  up?  (6  min.) 

9.  The. magnetic  circuit  of  a  dynamo  is  made  up  as  follows: 
cast-iron  ring,  length  of  path  25  cm.,  density  6000,  two  wrought- 
iron  cores  each  10  cm.  long,  with  density  of  16,000,    two  gaps 
5  mm.  long,  with  mean  density  of  8000,   the  path  through  the 
laminated  armature  15  cm.  long  with  a  density  of  10,000.     Re- 
quired the  ampere-turns  necessary  for  the  excitation.     (4  min.) 

10.  Given  a  cast-steel  ring,  8  inches  inside  diameter,  2  inches 
diameter  of  iron;  B  is  15,000  gausses.     Required  the  maxwells,  the 
oersteds  and  the  exciting  current  if  there  are  150  turns.     (8  min.) 

11.  Given  a  cast-iron  ring  100  cm.  in  mean  diameter.     Required 
the  number  of  amperes  necessary  to  make  B  equal  to  4000,  6000, 
8000  and  10,000  if  the  winding  be  of  854  turns.     (6  min.) 

12.  In  the  case  of  the  last  example,  if  the  winding  has  a  resist- 
ance of  3.76  ohms,  what  will  be  the  power  necessary  to  produce 
each  density?     Note  great  increase  in  power  with  increase  in 
density.     (5  min.) 

13.  In  order  to  raise  a  certain  weight  the  density  in  a  horse- 
shoe-shaped forged  hoisting  magnet  must  be  16,000  gausses;   the 
length  of  the  iron  circuit  is  50  cm.     If  there  are  1428  turns  carry- 
ing two  amperes,  how  near  to  the  weight  must  the  magnet  be 
brought  to  lift  it.     If  the  current  were  increased  to  7.14  amperes, 
how  many  turns  would  be  necessary?     (7  min.) 

14.  A  dynamo  magnetic  circuit  is  made  up  of  the  following 
parts:  27  cm.  in  the  armature  with  a  density  7000;  two  clearances 
of  3.4  mm.  each,  with  B  equal  to  12,000,  and  a  cast-steel  field  of 
53  cm.  with  a  density  of  15,000.     If  the  magnetizing  current  be 
10  amperes,  how  many  turns  will  be  necessary?     (4  min.) 

15.  A  2-kw.  transformer  has  a  magnetic  circuit  whose  section 
is  7.57  X  5.22  inches  and  length  22.3  inches.     If  there  are  800  turns 
and  a  maximum  density  of  5000  is  necessary,  what  will  be  the 
maximum  magnetizing  current?     (4  min.) 

16.  The  hysteresis  loop  for  a  sample  of  sheet  steel  is  carried 
to  13,750  gausses.     The  coordinates  are  chosen  so  that  one  inch 
represents  ten  units  of  magnetizing  force  and   10,000  gausses. 
The  area  of  the  curve  is  found  to  be  0.513  sq.  in.     What  would  be 


MAGNETISM  AND  MAGNETIC  CIRCUITS  13 

the  watts  loss  if  1000  cubic  centimeters  of  this  iron  were  subjected 
to  25  cycles  per  second?     (12  min.) 
Note.  —  1  watt-second  =  107  ergs. 

17.  Plot  the  curves  of  permeability  with  gausses  as  abscissae 
for  sheet  steel  and  for  cast  iron.     Take  the  data  from  the  tables 
on  page  105  and  plot  about  15  points  on  each.     (20  min.) 

18.  It  is  required  to  design  a  forged-steel  horseshoe  to  raise 
1000  pounds.     The  length  of  the  iron  path  in  the  magnet  is  50  cm. 
and  there  will  be  two  gaps  of  0.1  mm.     A  density  of  16,000  is  to 
be  used  in  the  steel,  which  on  account  of  leakage  will  be  reduced 
to  15,000  in  the  gaps.     On  account  of  greater  cross-section,  neglect 
the  reluctance  of  the  armature.     Required  the  diameter  of  the 
iron  and  the  gilberts  needed.     If  a  current  of  2  amperes  is  to  be 
used  for  excitation,  how  many  turns  will  be  required?     (9  min.) 


, 
Note.  —  The  magnetic  pull  is  -  —  grams  or  -—  —  •=  where 

4  7T   /\   yoJ.  J.J..J./ 

the  pull  is  in  pounds,  the  density  in  kilogausses  and  the  area 
(total  surface)  in  square  centimeters. 

19.  Two  parallel  surfaces  9X8  and  7  X  5  cm.  respectively  are 
10  cm.  (normal  to  surfaces)  apart.     Find  the  permeance  of  the 
space  between  them.     (1  min.) 

20.  Required   the  permeance   between   two   parallel   surfaces 
13.3  X  21.4    and   9.42  X  21.4    cm.    respectively,    and    6.31    cm. 
(normal  to  surfaces)  apart;  also  the  flux  if  the  winding  consists 
of  1270  turns  carrying  1.32  amperes.     (3  min.) 

21.  Two  surfaces  9  X  20  cm.  lie  in  the  same  plane  and  have 
their  long  edges  parallel  and  2  cm.  apart.     Required  the  per- 
meance of  the  path  between  them.     If  a  magnetomotive  force, 
produced  by  1000  turns,  carrying  two  amperes,  acted  between 
these  two  surfaces,  required  the  total  flux  between  them.     Assume 
the  lines  of  force  to  be  semicircles.     (3  min.) 

Note.   P  =  -log,^-2  =  0.733  L  logio^  where  L  =  length  of 

7T  U\  D\ 

parallel  edges  and  D2  and  A  =  the  distances  between  the  outer  and 
inner  edges.  (Found  by  integration  of  elementary  leakage  path. 
See  Standard  Handbook,  Sec.  5,  §  57.) 

22.  Two  surfaces  7.12  X  22.5  cm.  in  the  same  plane,  and  1.2 
cm.  apart  with  the  long  sides  parallel.     Assuming  the  lines  of  force 
to  be  semicircles;  required  the  permeance.     (5  min.) 


14  ELECTRICAL  ENGINEERING  PROBLEMS 

23.  The  flux  in  the  armature  of  a  bipolar,   110-volt,   6-kw. 
generator  is  3  X  106,  while  that  in  the  field  is  3.9  X  106.     Required 
the  leakage  coefficient.     (J  ram.) 

24.  In  a  bipolar,  110-volt,  4  kilowatt  generator,  the  armature 
density   is   14,000   and  the   total   flux  2.8  X  106.     The   leakage 
coefficient  is  1 .25 ;  required  the  area  of  the  field  magnet,  if  a  density 
of  12,000  be  used.     (3  min.) 

25.  In    a    50-kw.,    110-volt,    4-pole    generator    the    armature 
density  is  14,000,  and  the  double  cross-sectional  area  of  the  ring, 
550  sq.  cm.     In  the  field-magnet  ring  the  density  is  12,000  and 
the  double  area  is  820  sq.  cm.;  required  the  leakage  coefficient. 
(2  min.) 

26.  Given  the  leakage  coefficient  1.32  and  the  armature  flux 
10.63  megamaxwells.     The  yoke  of  the  dynamo  is  to  be  of  cast 
iron.     Required  the  dimensions  of  its  cross-section  in  inches  for 
B  equal  to  7200  if  its  length  is  to  be  twice  its  breadth.     (4  min.) 


CHAPTER  VI 
MAGNET    WINDINGS   AND   MAGNETS 

1.  Given  the  following  data:  resistivity  12;  ampere-turns  4000; 
mean  length  of  turn  18  inches;  e.m.f.  50  volts.     What  area  of  wire 
should  be  used  and  what  B.  &  S.  number  will  answer?     (2  min.) 

Note.  —  The  area  is  obtained  from  the  formula  for  drop  in 
terms  of  number  and  mean  length  of  turns,  resistivity,  current 
and  area. 

2.  A  magnet  requiring  20,000  ampere-turns  is  used  on  a  500- 
volt  circuit;  the  mean  length  of  a  turn  is  2  feet;  the  resistivity  is 
12;    the  circular  mils  per  ampere  1000.     Required  the  area  and 
the  B.  &  S.  number  of  the  wire  and  the  number  of  turns,  using  the 
B.  &  S.  size,  and  remembering  that  the  larger  wire  will  mean  more 
ampere-turns.     (5  min.) 

3.  Required  the  diameter  in  centimeters  of  a  horseshoe  magnet 
to  raise  4000  pounds  with  B  equal  to  16,000.     (4  min.) 

4.  Given  a  coil  of  wire  5J  inches  inside  and  6J  outside  diameter. 
13,400  ampere-turns  are  necessary;  resistivity  is  12.5;  e.m.f.  is 
125;  find  the  area  of  wire  necessary  and  the  B.  &  S.  number. 
(3  min.) 

5.  A  magnet  requires  10,000  ampere-turns  for  its  excitation. 
If  the  mean  length  of  a  turn  be  15  inches,  and  it  be  run  on  a  110- 
volt  circuit,  required  the  area  and  the  B.  &  S.  number  of  the  wire, 
resistivity  to  be  taken  as  12.     (3  min.) 

6.  What  must  be  the  cross-sectional  area  in  sq.  cm.  of  a  bar 
used  in  a  horseshoe  magnet  to  raise  500  pounds,  taking  B  as 
10,000?     (2  min.) 

7.  A  magnet  with  a  round  core  5  inches  in  diameter  has  560 
turns  of  No.  15  wire  wound  1  inch  deep.     How  many  feet  more  of 
wire  would  be  necessary  if  the  core,  having  the  same  area,  were 
rectangular  with  one  side  double  the  other.     What  per  cent  must 
the  area  of  the  wire  be  increased  if  the  work  of  excitation  is  to  be 
the  same?     (6  min.) 

8.  A  magnetic  circuit  consists  of  100  cm.  of  forged  steel  and 
£  cm.  of  air,  with  B  equal  to  16,000.     Required  the  ampere-turns 

15 


16  ELECTRICAL  ENGINEERING  PROBLEMS 

necessary;  also  on  a  110- volt  circuit,  the  area  and  the  B.  &  S.  size  of 
wire,  if  the  mean  length  of  turn  is  12  inches,  resistivity  12  and  if  1000 
circular  mils  per  ampere  be  used;  also  the  current  and  number  of 
turns.  (6  min.) 

9.  A  1-kw.,  2-pole,  110-volt  generator  (1800  r.p.m.)  has  the 
following  data  for  its  magnetic  circuit:    Length  of  two  air  gaps, 
0.6  cm.;    density,  4400;    length  in  armature,   13  cm.;    density, 
10,000;  length  in  cast-iron  field  magnet,  80  cm.;  area,  200  sq.  cm.; 
total  flux  in  armature,  1  X  106;  leakage  coefficient,  1.4;  power  to 
excite  the  field,  12%;  mean  length  of  a  turn,  24  inches.     Required 
the  ampere-turns,  size  of  wire  and  the  number  of  turns  necessary, 
taking  resistivity  as  12.5.     (5  min.) 

10.  Given  a  round  bar  3  inches  in  diameter,  bent  into  a  magnet, 
and  having  the  length  of  magnetic  circuit  15.75  inches.     Find  the 
number  of  turns  necessary  to  give  a  pull  of  2433  pounds  with  a 
current  of  5.27  amperes.     (10  min.) 

11.  Required  to  design  a  magnet  for  use  on  a  100-volt  circuit, 
and  to  raise  2000  pounds.     Resistivity  is  12.     Allow  1000  circular 
mils  per  ampere.     B  =  16,000.     Use  double  cotton  covered  wire 
(see  table,  page  104).     It  is  3J  inches  between  the  insides  of  the 
shanks  of  the  magnet,  which  are  4  inches  long  and  are  joined  by  a 
semicircular  yoke;  allow  J  mm.  air  gap  for  the  roughness  of  the 
iron  piece  to  be  raised,  and  1  inch  for  the  thickness  of  the  winding. 
Find  the  diameter  of  the  bar,  area  and  B.  &  S.  number  of  the 
wire,  the  number  of  turns  and  the  current.     (30  min.) 

12.  Design  a  magnet  for  use  on  a  110-volt  circuit,  to  raise  2| 
tons  with  a  factor  of  safety  of  1 J.     Length  of  the  magnetic  circuit 
in  iron,  50  cm.,  and  in  air  at  joints,  ^  cm.     Allow  1000  circular 
mils  per  ampere;  the  winding  is  1  cm.  deep;  use  cast  steel  with 
16   kilogausses,    and   take  the  resistivity  as  12.     Required  the 
number  of  ampere-turns  necessary  for  the  iron;   the  number  of 
ampere-turns  necessary  for  the  air;  diameter  of  the  iron;  and  the 
area,  size  and  total  number  of  turns  of  wire.     (25  min.) 

13.  A  440-kw.,  10-pole,  500-volt  (at  no  load),  85-r.p.m.  genera- 
tor has  the  following  data  for  its  magnetic  circuit:  Cast-iron 
yoke  (TV  of  the  ring),  130  cm.  in  length  with  a  density,  7500; 
laminated  cores,  45  cm.  each  in  length  with  a  density,  13,500; 
armature  body  (T^  of  the  ring),  85  cm.  in  length;  with  a  density, 
9000;  teeth,  4.5  cm.  in  length;  with  a  density,  19,000;  air  gaps, 
0.50  cm.  in  length;  with  a  density,  16,000.     The  mean  length  of 


MAGNET  WINDINGS  AND  MAGNETS       17 

the  shunt  turn  is  79  inches  and  the  total  power  taken  by  the 
shunt  at  no  load  is  1.34%  of  the  output  of  the  machine.  Required 
the  ampere-turns,  the  proper  area  of  the  shunt  wire  and  the  nearest 
B.  &  S.  size;  also  the  number  of  turns;  take  the  resistivity  as  12.5. 
(16  min.) 

14.  It  is  desired  to  build  a  magnet  of  rectangular  cross-section, 
width  equal  to  twice  the  breadth,  and  to  be  made  of  cast  iron. 
Length  of  gap,  f  inch;  length  of  path  in  the  iron,  35  inches;  total 
induction,  3  megamaxwells;  winding,  2  inches  deep;   density  hi 
gap,  7000.     With  wire  at  20  cents  per  pound  and  iron  at  2|  cents, 
what  is  the  cost  of  material,  arid  will  it  be  cheaper  to  use  a  density 
of  6000  or  of  7000,  and  how  much?     Use  No.  12  wire  which  runs 
20.5  pounds  per  1000  feet,  and  an  exciting  current  of  10  amperes. 
Take  the  volume  of  the  iron  as  area  times  length,  and  its  weight  as 
0.26  pounds  per  cu.  in.     (20  min.) 

15.  An  iron-clad  magnet,  to  be  used  as  a  clutch,  has  a  central 
pole  10  cm.  in  diameter,  and  a  mean  diameter  of  outside  ring  of 
20  cm.     Required  the  thickness  of  the  cylindrical  pole  and  the 
horse-power  that  the  clutch  will  transmit  at  500  r.p.m.  if  B  be 
taken  as  16,000,  and  if  the  normal  pull  is  to  be  taken  as  ten  times 
the  tangential  pull.     Divide  the  central  pole  into  a  circle  2  cm.  in 
diameter,  surrounded  by  two  rings  each  2  cm.  wide;    take  the 
torque  as  made  up  of  that  due  to  each  of  these  taken  at  its  mean 
radius,  plus  that  of  the  outside  ring  at  its  mean  radius.     (15  min.) 


CHAPTER  VII 

GENERATION    OF   ELECTROMOTIVE   FORCE 
ARMATURE  DROP 

1.  Required  the  e.m.f.  in  volts  of  a  conductor  18  cm.  long, 
cutting  a  field  of  B,  9000,  at  a  velocity  of  1200  meters  per  minute. 
(J  min.) 

2.  A  conductor  on  the  surface  of  an  armature  is  50  cm.  long 
and  cuts  through  the  magnetic  field  of  density  9000  under  a  pole 
piece;  the  speed  is  27  meters  per  second.     What  e.m.f.  will  be  set 
up  in  it?     (1  min.) 

3.  A  rectangular  coil  having  85  turns,  20  X  40  cm.,  is  moved 
in  the  direction  of  its  long  axis  through  a  magnetic  field  at  the  rate 
of  12  meters  per  second.     One  of  its  short  sides  is  outside  of  the 
field.     The  terminals  of  the  coil  are  connected  to  an  oscillograph 
which  shows  that  the  e.m.f.  set  up  is  0.012  volts.     What  is  the 
density  of  the  field?     (1  min.) 

4.  If  the  coil  of  the  last  problem  were  moved  in  a  direction 
making  30  degrees  with  the  axis  of  a  field  of  density  8000,  what 
would  be  the  e.m.f.  set  up?     (1  min.) 

5.  A  1-kw.,  2-pole  generator  has  228  conductors  and  a  speed  of 
2000  r.p.m.    The  flux  is  1.6  X  106  maxwells;  what  is  the  e.m.f.? 
(1  min.) 

6.  A  2-pole,   8-kw.,   125-volt  generator  has   180  conductors. 
The  flux  is  2.5  X  106.     At  what  r.p.m.  must  the  generator  be  run? 
(1  min.) 

7.  A  1000-kw.,  16-pole,  500-volt  generator  is  to  run  at  90  r.p.m. 
There  are  2304  conductors.     What  must  be  the  flux  per  pole,  and 
what  the  area  of  the  pole  pieces  with  an  air-gap  density  of  8600? 
(2  min.) 

8.  The  copper  loss  in  the  armature  of  problem  7  is  24  kw. 
What  increased  voltage  would  the  machine  generate  at  full  load  in 
order  to  give  500  volts,  at  the  terminals,  and  what  per  cent  increase 
in  the  flux  would  be  necessary?     (1  min.) 

9.  There  are  100  complete  turns  on  a  bipolar  armature  which 

18 


ARMATURE  DROP  19 

makes  1500  r.p.m.;    the  flux  is  2  X  106.     Required  the  current 
if  the  total  resistance  is  3  ohms.     (1  min.) 

10.  A  series  machine  has  a  resistance  of  0.2  ohm.      For  5 
amperes  the  terminal  e.m.f.  is  99  volts;  increasing  the  current  to 
10  amperes  increases  B  from  10,000  to  15,000;  required  the  e.m.f . 
at  the  terminals  of  the  machine  for  a  current  of  10  amperes. 
(1  min.) 

11.  An  armature  core  is  15  cm.  in  diameter;  the  shaft  is  3  cm. 
in  diameter;  the  net  iron  armature  length  is  15  cm.;  the  armature 
density  is  12,000  and  there  are  100  conductors;  required  the  e.m.f. 
at  2400  r.p.m.     (2  min.) 

12.  A  dynamo  has  64  conductors;   the  length  of  the  armature 
is  30  cm.  and  its  diameter  is  25  cm.     The  pole  pieces  cover  three- 
fourths  of  the  armature,  and  the  density  in  the  air  gap  is  10,000. 
Required  the  e.m.f.  at  1000  r.p.m.     (2  min.) 

13.  A  50-kw.,  4-pole,   120-volt  machine,  designed  to  run  at 
680  r.p.m.,  is  to  have  its  speed  reduced  to  340  to  permit  of  direct 
coupling  to  an  engine.     Its  armature  winding  is  made  up  of  128 
bars,  each  5  X  10  mm.     What  will  be  the  values  of  the  e.m.f., 
current  capacity  and  power  after  this  speed  reduction?     State 
what  change  must  be  made  in  the  winding  to  bring  the  e.m.f.  back 
to  120  volts,  and  what  will  then  be  the  current  capacity  and  power. 
The  winding  is  to  occupy  the  same  space  in  both  cases,  neglecting 
insulation.     (3  min.) 

14.  In  the  machine  of  problem  9,  Chap.  VI,  it  is  required  to 
increase  the  e.m.f.  at  full  load  to  135  volts.     What  per  cent  increase 
must  be  made  in  the  flux  in  order  to  accomplish  this?     (1  min.) 

15.  A  250- volt,  40-kw.,  4-pole  generator  has  a  4-path  winding 
of  272  conductors.     By  leaving  out  two  of  these  conductors  this 
winding  may  be  changed  to  a  2-path  winding.     What  would  then 
be  the  e.m.f.,  current  capacity  and  power  of  the  machine?     How 
much  should  the  speed  be  increased  to  make  up  for  the  loss  of  the 
two  conductors?     (2  min.) 

16.  A  150-kw.,  6-pole,  250-volt,  450-r.p.m.  generator  has  496 
armature  conductors,  wound  6-path.     If  this  were  changed  to  a 
2-path  winding,  what  would  its  e.m.f.,  current  capacity  and  power 
rating  become?    What  would  be  the  e.m.f.  generated  in  one  con- 
ductor?    (2  min.) 

17.  The  resistance  of  the  6-path  armature,  including  brush 
drop  of  problem  16,  is  0.0081  of  the  series  coil  0.00083,  and  of  the 


20  ELECTRICAL  ENGINEERING  PROBLEMS 

shunt  field  61  ohms.  What  e.m.f.  will  have  to  be  generated  at 
full  load  in  order  to  get  the  250  volts  at  the  terminals?  If  the 
flux  were  kept  constant  in  this  machine  from  no  load  to  full  load, 
what  would  be  the  regulation  expressed  in  per  cent  at  the  rated 
voltage  at  full  load?  By  what  per  cent  must  the  flux  density  be 
increased  to  maintain  a  constant  voltage?  (2  min.) 

18.  The  armature  of  a  30-kw.,   110-volt,  4-pole  machine  is 
wound  multiplex  with  two  bar  windings  of  156  conductors  each. 
By  changing  the  method  of  connection  to  the  commutator  and 
without  other  change  this  may  be  made  a  single  winding.     What 
will  then  be  the  e.m.f.,  current  capacity  and  power  rating  of  the 
armature?     (1  min.) 

19.  A  maker  uses  the  same  armature  core  for  125  and  500- volt 
machines  of  the  same  watt  capacity.     At  125  volts  there  are  144 
turns  of  No.  8  wire.     Specify  the  number  of  turns  and  the  B.  &  S.. 
number  for  the  500- volt  machine  of  the  same  speed.     The  winding 
is  to  occupy  the  same  space  neglecting  additional  space  taken  by 
the  insulation.     (1  min.) 

20.  A  4-path  armature  has  42  slots  with  wire  space  J  X  \  inch, 
and  each  slot  contains  8  No.  8  wires.     The  e.m.f.  is  120  volts. 
This  armature  is  rewound  with  No.  14  wire,  putting  in  as  many 
conductors  in  a  slot  as  possible.     Required  the  e.m.f.  in  the  second 
case  and  the  power  in  both  cases,  using  600  circular  mils  per 
ampere.     The  speed  remains  constant.     (4  min.) 

21.  If  a  unipolar  dynamo  runs  at  6000  r.p.m.  and  the  strength 
of  field  is  16,000  gausses,  what  must  be  the  diameter  of  the  disk  to 
generate  5  volts,  the  inside  diameter  being  4  inches.     (3  min.) 

22.  A  certain  110-volt  shunt  generator  when  run  at  0.8  normal 
speed  gives  65  volts.     What  percentage  change  has  taken  place 
in  the  flux?     When  run  at  20%  above  normal  speed  the  voltage 
is  145;  what  is  the  change  in  the  flux?     (4  min.) 

23.  A  4-pole  generator  has  a  speed  of  1500;   the  length  of  the 
armature  is  15  cm.  and  its  diameter  is  25  cm.     The  pole  covers 
60  degrees.     The  armature  is  4-path  and  has  31  slots  each  con- 
taining 8  conductors  of  No.  10  wire.     The  gap  density  is  10,000. 
Required  the  e.m.f.  generated  and  the  output  in  kilowatts  with 
400  circular  mils  per  ampere  allowed  in  the  armature  winding. 
(4  min.) 


CHAPTER  VIII 
ARMATURE    WINDINGS 

The  three  usual  methods  of  depicting  an  armature  winding  are 
by  the  winding  table,  the  development  and  the  radial  or  end-view 
diagram.  Give  all  of  these  for  each  of  the  following  cases,  as 
assigned.  Also  show  the  location  of  the  poles  and  brushes  in  each 
case.  Coordinate  paper  is  convenient  for  making  the  develop- 
ments. 


Prob- 
lem. 

Polea 

Paths 

Con- 
ductors 

Right  or 
left  hand 

Com. 
Seg- 
ments. 

Type 

Time 

1 

2 

26 

Right 

13 

Simplex 

10  min. 

2 

2 

36 

Left 

18 

Simplex 

10  min. 

3 

2 

72 

Right 

18 

Simplex 

10  min. 

4 
5 

4 
4 

4 

2 

36 
26 

Right 
Left 

18 
13 

Simplex 
Simplex 

10  min. 
10  min. 

6 

4 

2 

52 

Right 

13 

Simplex 

10  min. 

7 

6 

6 

36 

Right 

18 

Simplex 

10  min. 

8 

6 

2 

40 

Left 

20 

Simplex 

10  min. 

9 

4 

4 

36 

Right 

20 

Duplex 

20  min. 

Note.  —  Armature  windings  may  be  studied  advantageously 
with  the  aid  of  small  wooden  models,  say  about  three  inches  in 
diameter  by  one  inch  long.  Upon  the  cylindrical  surface  of  these 
saw  cuts  are  made  to  represent  the  slots.  On  the  front  end  of  this 
model  may  be  a  boss  one  inch  in  diameter  and  about  J  inch  long, 
to  represent  the  commutator,  with  small  brads  driven  in  opposite 
alternate  slots  to  mark  the  segments  and  serve  for  the  attachment 
of  a  winding  of  thread.  It  is  also  well  to  have  a  small  boss  on  the 
back  side  to  represent  the  shaft.  The  following  problems  are 
based  upon  such  model  armatures,  having  26  and  36  slots  respec- 
tively: 

10.  Put  the  winding  of  problem  1  on  the  proper  model  using 
thread.  Take  this  off  and  wind  on  the  corresponding  left-hand 
winding.  (8  min.) 

21 


22  ELECTRICAL  ENGINEERING  PROBLEMS 

11.  Same  as  problem  10  except  that  the  winding  of  problem  2 
is  to  be  used.     (8  min.) 

12.  Same  as  problem  10  except  that  the  winding  of  problem  4 
is  to  be  used.     (8  min.) 

13.  Same  as  problem  10  except  that  the  winding  of  problem  5 
is  to  be  used.     (8  min.) 

14.  Same  as  problem  10  except  that  the  winding  of  problem  7 
is  to  be  used.     (8  min.) 

15.  Same  as  problem  10  except  that  the  winding  of  problem  9 
is  to  be  used.     (10  min.) 


CHAPTER  IX 

ARMATURE-CIRCUIT    CALCULATIONS,    RESISTANCES,    CURRENT 

CAPACITY,    ETC. 

1.  In  a  certain  dynamo  the  armature  resistance  is  0.1  ohm; 
when  giving  50  amperes  it  actually  generates  105  volts;   what  is 
the  e.m.f.  at  the  terminals?     (f  min.) 

2.  A  bipolar  machine  having  an  armature  with  48  segments 
generates  at  full  load  an  actual  e.m.f.  of  115  volts;  the  resistance 
(brush  to  brush)  is  0.037;  with  93  amperes  delivered  by  the  arma- 
ture, what  will  be  the  average  difference  of  potential  per  segment? 
(1  min.) 

3.  In  the  above  problem  what  is  the  actual  resistance  between 
two  adjacent  commutator  bars?     (J  min.) 

4.  A  2-pole  armature  is  wound  with  550  turns,  each  including 
three  feet  of  No.  10  wire.     Required  the  resistance  from  brush  to 
brush  at  70°  C.  (for  resistivity  see  table  on  page  104).     Also  find 
the  current  output  of  the  armature  at  600  circular  mils  per  ampere. 
Also  the  armature  drop  with  this  current.     (2  min.) 

5.  Required  the  resistance  and  the  current  output,  if  the  above 
armature    be    connected  for    a   4-pole,   4-path    (parallel-wound) 
machine.     Also  for  a  6-pole,  2-path  (series  wound).     Also  for  a 
10-pole,  10-path  machine.     (4  min.) 

6.  A  2-path  armature  is  wound  duplex  for  250  volts.     Each 
winding  is  made  up  of  199  turns  of  4  feet  each  of  No.  12  wire. 
For  500  volts  the  coils  of  the  two  windings  are  simply  put  in  series 
by  changing  the  connections  at  the  commutator  segments.     Re- 
quired the  resistances  in  both  cases  for  40°  C.     For   the  same 
current  density  how  does  the  power  lost  in  the  armature  winding 
in  the  two  cases  compare?     (3  min.) 

7.  If  there  is  a  difference  of  3  volts  in  the  e.m.f. 's  set  up  under 
the  poles  in  a  2-pole  generator,  the  armature  resistance  being 
0.2  ohm  and  the  total  current  through  the  two  sides  being  27 
amperes,  what  current  will  flow  under  each  pole?     What  current 
will  flow  when  the  outside  circuit  is  open?     (2  min.) 

23 


24      ELECTRICAL  ENGINEERING  PROBLEMS 

8.  By  test  a  bipolar  shunt  generator  gives  90  volts  and  25 
amperes  to  the  outside  circuit;  if  the  resistance  of  the  armature 
be  0.324  ohms,  the  resistance  of  the  field  be  41.0  ohms,  required 
the  current  through  the  armature  conductor  and  the  B.  &  S.  size 
if  500  circular  mils  per  ampere  be  used.  Also  the  total  e.m.f. 
generated  by  the  armature.  (2  min.) 


CHAPTER  X 
ARMATURE    REACTIONS 

1.  A  bipolar  armature  has  420  conductors,  and  its  field  has  a 
polar  angle  of  120  degrees.     When  it  is  giving  100  amperes  from 
the  brushes,  how  many  cross  and  how  many  back  ampere-turns 
will  be  present,  supposing  that  the  brushes  are  placed  opposite 
the  pole  tips?     (1  min.) 

2.  The  pole  pieces  of  a  bipolar  generator  subtend  108  degrees 
each,  and  the  armature  has  63  slots  with  4  conductors  each.     If 
the  brushes  are  opposite  the  pole  tips,  how  many  cross  and  how 
many  back  turns  will  be  present?     If  the  machine  is  giving  40 
amperes,  how  many  cross  and  back  ampere-turns  will  there  be? 
(1  min.) 

3.  A  4-pole,  55-kw.,  125-volt,  4-path  (parallel- wound)  generator 
has  poles  subtending  60  degrees  each.     There  are  100  slots  with 
two  conductors  each.     What  will  be  the  back  and  cross  ampere- 
turns  when  the  machine  is  giving  its  full  load  if  the  brushes  are 
opposite  the  pole  tips?     If  the  connections  of  this  armature  be 
changed  so  as  to  use  it  in  a  2-pole  field  having  the  same  total  area 
of  pole  face,  what  will  the  cross  and  back  ampere-turns  become? 
The  same  current  density  is  used  in  the  conductors.     (2  min.) 

4.  A  100-kw.,  250-volt,  8-pole  generator  has  an  armature  cir- 
cumference of  360  cm.     The  width  of  each  pole  face  is  36  cm., 
there  are  167  slots  with  two  conductors  each,  and  they  are  con- 
nected as  a  2-path  (series)  winding.     How  many  cross  and  back 
ampere-turns  are  present  at  full  load  with  the  brushes  opposite  the 
pole  tips?     How  many  would  there  be  if  the  same  conductors  used 
at  the  same  current  density  could  be  connected  as  an  8-path 
(parallel)   winding?     How  many  if  the  same   conductors  were 
reconnected  and  used  in  a  2-pole  field  with  the  same  total  polar 
area?     (2  min.) 

6.  If  in  problem  1  the  gap  density  be  8000  and  the  density  due 
to  the  cross  field  must  not  exceed  75%  of  the  gap  density,  what 
will  be  the  smallest  permissible  length  of  gap?  How  much  could 
this  be  reduced  if  the  machine  were  4-pole?  (1  min.) 

25 


26      ELECTRICAL  ENGINEERING  PROBLEMS 

6.  If  the  flux  be  3  X  106,  the  reluctance  of  the  whole  circuit  be 
0.001  and  there  be  ten  back  turns,  required  the  total  number  of 
ampere-turns  on  the  field  necessary  if  the  armature  be  giving  48.6 
amperes.     (1  min.) 

7.  A  bipolar  smooth-core  machine  has  an  air  gap  of  f  inch,  a 
gap  density  of  12,000,  a  polar  angle  of  135  degrees  and  200  con- 
ductors.    How  much  current  could  be  taken  from  the  machine 
without  reversing  the  field  under  the  pole  tips?     (4  min.) 

Note.  —  On  account  of  their  greater  simplicity  most  of  the 
problems  involving  length  of  air  gap  are  given  as  applying  to  the 
old-style  smooth-core  machines.  The  flux  conditions  in  the  gap 
and  teeth  of  a  slotted  core  dynamo  are  too  complicated  to  lend 
themselves  readily  to  problems  of  this  kind.  The  principals 
involved  are,  however,  similar. 

8.  Given  the  following  data  of  a  2-pole  machine:  the  number  of 
conductors  is  320,  the  gap  density  is  6000;  the  length  of  gap  (two 
sides)  is  1.6  cm.;   80%  of  the  surface  of  the  armature  is  covered 
by  the  poles.     If  the  cross  ampere-turns  are  not  to  exceed  0.6  of 
the  part  of  the  field  ampere-turns  applied  in  the  gap,  how  much 
current  can  the  above  armature  carry?     How  much  if  only  75% 
of  the  surface  of  the  armature  be  covered  by  the  poles?     (3  min.) 

9.  A  110-volt  shunt  dynamo  has  1130  field  turns  with  a  shunt 
resistance  of  55  ohms.     There  are  120  armature  conductors  and 
a  20-degree  angle  of  lead.     The  polar  angle  is  120  degrees.     The 
machine  is  feeding  132  15-watt  lamps.     Required  the  cross,  back 
and  effective  ampere-turns.     (3  min.) 

10.  Determine  the  number  of  cross  ampere-turns  per  pole,  back 
ampere-turns  per  pole  and  series  turns  per  pole  to  balance  the 
back  turns  in  the  case  of  a  dynamo  having  the  following  data: 
Conductors  500,  total  current  200,  4-pole,  2-path  winding,  polar 
angle  60  degrees,  angle  of  lead  10  degrees,  0.9  of  the  total  current 
flowing  through  the  series  turns;  leakage  coefficient,  1.3.     (4  min.) 

11.  Given  a  4-pole  smooth-core  dynamo,  with  the  polar  angle 
60  degrees,  angle  of  lead  12  degrees,  240  conductors,  current  (total) 
824,  4-path  winding,  leakage  coefficient  1.25;  required  the  series 
turns  per  pole  necessary  to  balance  the  back  turns;    also  the 
minimum  thickness,  in  inches,  of  the  air  gap  without  possible 
reversal  of  the  field  under  the  pole  tip,  if  the  gap  density  be  10,000. 
If  this  machine  were  bipolar  with  f  the  total  current  and  double 
the  polar  angle,  what  would  the  above  quantities  be?     (4  min.) 


ARMATURE   REACTIONS  27 

12.  A  2-pole  armature  has  64  segments,  two  turns  per  segment. 
The  total  current  is  120;  angle  of  lead  25  degrees;  brushes  opposite 
pole  tips.     The  gap  density  is  10,000  and  the  leakage  coefficient, 
1.35.     Required:  (a)  The  cross  ampere-turns.     (6)  The  back  am- 
pere-turns,    (c)  The  series  turns  necessary  to  balance  the  back 
ampere-turns  if  0.8  of  the  total  current  be  used  in  the  series  wind- 
ing,    (d)   The  least  thickness  of  air  gap  that  could  be  used  without 
reversing  the  field  under  the  pole  tip  (cm.),     (e)  The  same  quan- 
tities if  the  polar  angle  be  increased  to  145  degrees.     (/)  Also  if, 
with  290  degrees  covered  by  the  pole  pieces,  the  machine  be  made 
a  4-pole  machine  with  a  2-path  winding,  find  values  for  each 
magnetic  circuit,     (g)  Also  the  gap  thickness  in  the  first  case  if 
the  gap  density  be  reduced  to  8000.     (12  min.) 

13.  In  a  smooth-core  machine,  if  the  density  due  to  the  field 
winding  be  7000  and  the  length  of  one  air  gap  0.8  cm.;  if  there  be 
260  conductors  and  the  polar  angle  be  125  degrees;  plot  the  curves 
of  density  across  the  pole  face  with  0,  50  and  200  amperes  given 
by  the  armature.     (12  min.} 

14.  If  in  the  machine  of  problem  3  the  slots  be  2.4  cm.  deep  and 
the  tooth  density  20,000,  also  if  the  clearance  be  2  mm.  and  the 
density  above  the  teeth  be  on  account  of  the  spreading  of  the  lines 
80%  of  the  tooth  density,  also  neglecting  all  other  conditions  than 
the  effect  of  reluctance  on  the  circuit  of  the  cross-magnetization 
(through  the  teeth  under  the  tips),  what  current  would  reduce  the 
density  under  the  leading  pole  tip  to  75%  of  its  no-load  value? 
(5  min.) 

Note.  —  The  effect  of  the  unequal  reluctance  under  the  two  tips 
would  effect  the  distribution  of  the  main  field,  thus  requiring  a  still 
larger  current  to  produce  the  proposed  change  in  density. 


CHAPTER  XI 
MAGNETIZATION   CURVES 

1.  A  4-kw.,  110-volt,  2-pole  generator  (ISOOr.p.m.)  with  a  cast- 
iron  field,  has  lengths  and  cross-sectional  areas  of  its  magnetic  path 
as  follows:    In  the  field,  110  cm.,  470  sq.  cm.;  2  gaps,  7  mm.,  540 
sq.  cm.;   armature,  20  cm.,  165  sq.  cm.     The  leakage  coefficient 
is  1.3.     Plot  magnetization  curves  for  the  gap,  the  armature,  the 
field  and  hence  for  the  whole  machine.     Plot  in  terms  of  total 
armature  flux  and  field  ampere-turns  obtaining  five  points  by 
steps  of  500,000  each.     Give  also  on  the  same  curves,  scales  of 
e.m.f.  and  field  current;  the  field  turns  being  2400  and  the  num- 
ber of  armature  conductors  216.     From  the  curve  determine  the 
armature  flux  at  no  load.     Find  also  the  field  currents  to  give  80, 
100  and  120  volts  at  no  load.     (40  min.) 

2.  A  150-kw.,  6-pole,  250-volt  generator,  450  r.p.m.  (same  as 
in  problem  6,  Chap.  XIV)  has  the  following  data  for  its  magnetic 
circuit:    Yoke,  cross-sectional  area,  284  sq.  cm.,  length,  66  cm.; 
cores,  area,  506  cm.,  length,  35.5  cm.  each;   armature,  area,  684 
sq.  cm.,  length,  38  cm.;    teeth,  total  area  under  pole  face,  413 
sq.  cm.,  length,  4.1  each;    gaps,  area  increased  on  account  of 
spreading  to  658  cm.,  length,  0.8  cm.  each;    leakage  coefficient, 
1.17.     Plot  magnetization  curves  for  the  field,  gaps  and  armature, 
and  from  these  for  the  whole  machine,  obtaining  four  points  for 
armature  fluxes  equal  1,  3,  5  and  7  X  106  lines.     If  there  are  1800 
field  turns  per  pole  and  a  total  of  496  armature  conductors,  wound 
6-path,  lay  off  scales  for  field  current  and  generated  e.m.f.     Find 
the  field  current  for  200,  250  and  275  volts  at  no  load.     The  field 
is  all  of  cast  steel.     (50  min.) 

3.  It  is  desired  to  use  the  machine  described  in  problem  1  over- 
compounded  so  that  at  full  load  it  will  generate  10%  more  voltage. 
What  per  cent  must  the  ampere-turns  be  increased  to  do  this? 
It  is  also  desired  to  use  this  same  machine  to  fill  a  special  order  for 
a  90-volt  machine  to  overcompound  10%  at  full  load.      What 
ampere-turns  will  be  needed  at  no  load,  and  by  what  per  cent  must 
they  be  increased  at  full  load?     (5  min.) 

28 


MAGNETIZATION  CURVES  29 

Note  the  difference  in  these  two  cases  due  to  the  shape  of  the 
curve. 

4.  Plot  a  curve  to  show  the  effect  on  the  magnetization  curve 
of  using  in  problem  1  a  toothed  core  armature  with  a  clearance  of 
0.3  cm.     Assume  the  teeth  to  be  1.5  cm.  long  and  to  occupy  50% 
of  the  area  under  the  pole  face.     (30  min.) 

5.  To  maintain  the  same  voltage  in  the  generator  of  problem 
2,  if  its  speed  were  to  be  decreased  to  425  r.p.m.,  what  change  in 
the  field  ampere-turns  would  be  necessary?     Also  if  the  speed 
were  to  be  increased  to  475  r.p.m.?     (5  min.} 

6.  The  machine  of  problem  2  has  an  armature  resistance  of 
0.0081  and  the  equivalent  demagnetizing  turns  on  the  armature 
are  28.     Determine  from  the  magnetization  curves  the  ampere- 
turns  and  the  field  current  necessary  to  operate  the  machine  at 
full-load  current  and  voltage.     Also  the  voltage  necessary  on  the 
field  to  give  this  field  current,  the  shunt-field  resistance  being 
61  ohms.     Could  the  generator  be  operated  as  a  plain  shunt 
machine?     (4  min.) 

7.  Determine  from  the  data  given  in  problem  6  the  number  of 
series  turns  needed  to  operate  this  machine  as  a  flat  compound  and 
also  as  a  10%  over-compound  generator.     (3  min.) 


CHAPTER  XII 
CHARACTERISTICS 

Note.  —  In  plotting  always  use  the  same  scale  for  abscissae  and 
ordinates. 

Unless  otherwise  stated  "armature  current"  means  the  com- 
bined current  of  all  the  parallel  paths,  and  " armature  resistance" 
means  the  resistance  from  brush  to  brush. 

1.  Assuming   that   a   certain  magneto   generates   a   constant 
«.m.f .  of  60  volts  regardless  of  load,  and  has  an  armature  resistance 
of  120  ohms,  what  current  will  it  give  when  short  circuited?     With 
what  current  will  it  give  50  volts  at  the  terminals?     (J  min.) 

2.  Required  the  e.m.f.  generated  and  the  current  in  the  arma- 
ture in  each  of  the  following  cases: 

(a)  A  100-ohm  magneto  sends  0.1  ampere  through  1000  ohms 
resistance. 

(6)  A  shunt  machine  having  field  and  armature  resistances,  50 
and  0.01  ohms,  supplies  50  100-volt  40-watt  lamps  connected  in 
parallel.  (1  min.) 

3.  Required  the  e.m.f.  generated  and  the  current  in  the  arma- 
ture in  the  two  following  cases: 

(a)  A  series  machine  supplies  ten  45-volt  10-ampere  lamps 
through  a  line  having  5  ohms  resistance;  the  field  and  armature 
resistances  are  0.2  and  0.3  ohms. 

(6)  A  long-shunt  compound  generator  gives  500  volts  and  50 
amperes;  the  armature  and  series  and  shunt-field  resistances  are 
0.03,  0.02  and  500  ohms.  (1  min.) 

4.  The  curve  of  generated  e.m.f.  with  current,  i.e.  the  total 
characteristic  of  a  separately  excited  generator,  is  given  by  the 
following  points: 

Current 0          20          40.        60 

E.m.f 110        109        107        104 

The  armature  resistance  is  0.05  ohm. 

Construct  the  above  curve  and  derive  the  external  characteristic. 

(5  min.) 

30 


CHARACTERISTICS  31 

6.  A  10-kw.,  long-shunt,  compound  generator  has  armature, 
shunt  and  series-field  resistances  0.05,  22  and  0.03  ohms.  It  is 
supplying  at  its  terminals  200  55-watt  110-volt  lamps.  Re- 
quired the  total  generated  e.m.f.  and  the  total  armature  current. 
(1  min.) 

6.  If  the  machine  of  problem  5  is  connected  as  a  short  shunt 
with  the  same  terminal  e.m.f.,  required  the  generated  e.m.f.  and 
current.     (1  min.) 

7.  If  the  generator  in  problem  5  has  a  regulating  " shunt"  of 
0.1  ohm  resistance,  in  parallel  with  its  series  turns,  how  will  the 
series  ampere-turns  be  changed?     (1  min.) 

8.  A  long-shunt,    compound,   80-kw.   generator   is   supplying 
through  a  feeder  having  0.2  ohm  resistance  the  following  load: 
A  220-volt  motor  taking  22  kw.  and  1100  16-c.p.,  220-volt  carbon 
lamps  taking  four  watts  per  candle  power.     The  armature  resist- 
ance is  0.02  ohms,  and  the  shunt  and  series-field  resistances  are  22 
and  0.01  ohms.     Required  the  armature  current  and  the  generated 
e.m.f.     (2  min.) 

9.  The  following  data  are  obtained  from  a  test  of  a  shunt 
machine : 

Volts 110     100    90    80    70    60    50    40 

Amperes 0      40    59     65     67     65     59     50 

The  armature  and  field  resistances  are  0.12  and  30  ohms.  Plot 
this  curve,  and  from  it  plot  the  total  characteristic.  (10  min.) 

10.  Given  the  following  data  for  a  series  external  characteristic : 

Volts 20    40    60    80     100     120     132     135     133     120 

Amperes 10     19    26    35      45      60      80     100     120     150 

RO  =  0.1  and  R/  =  0.05.  Plot  this  curve,  and  from  it  plot 
the  total  characteristic.  Also  plot  the  latter  curve  if  the  speed  be 
increased  20%.  (10  min.) 

11.  Given  these  data  for  a  shunt  total  characteristic: 

Volts 143     140     136     130     120     110    90 

Amperes 10      22      40      60      70      75    78 

Plot  this  curve  and  from  it  plot  the  external  characteristic, 
having  given  Ra  =  0.08,  R,  =  30.  (10  min.) 

12.  Given  the  following  data  of  an  external  shunt  characteristic : 

Volts 216    207     197     187     175     161     146 

Amperes 0        4        8       12      16      20      24 


32      ELECTRICAL  ENGINEERING  PROBLEMS 

R«  =  0.4  and  Ra  =  80.     Plot  this  curve  and  from  it  the  total 
characteristic.     (10  ram.) 

13.  Plot  the  external  straight-line  characteristic  of  a  550-volt 
(at  full  load),  110-kw.,  compound- wound,  short-shunt  generator, 
over-compounded  10%.     If  Ra  =  0.09,  R/  =  0.04,  R,  =  100,  plot 
also  the  total  characteristic.     (10  min.) 

14.  A  450-kw.,  compound  generator,  long  shunt  (85  r.p.m.), 
gives  500  volts  on  no  load  and  is  over-compounded  10%.     R0  = 
0.018,  R/  =  0.0055.     The  full-load  shunt  current  is  10.5  amperes. 
Construct  the  external  and  total  characteristics.     (10  min.) 


CHAPTER  XIII 
HEATING   AND    RATED    CAPACITY   OF   DYNAMOS 

1.  If  a  cylindrical  coil  6  inches  in  diameter  and  10  inches  long 
has  a  resistance  of  8  ohms,  and  a  current  of  5  amperes  is  passed 
through  it,  what  will  be  the  final  temperature,  if  there  be  radiated 
0.01  watt  per  sq.  in.  per  degree  F.?     (2  mm.) 

2.  If  the  coil  of  a  magnet  with  the  same  character  of  surface 
as  above  were  4  inches  in  diameter  outside  and  5  inches  long,  how 
many  watts  could  be  allowed  in  the  same,  if  its  maximum  tem- 
perature above  the  atmosphere  is  to  be  80  degrees?     Figure  only 
the  cylindrical  surface  as  radiating.     If  operated  on  a  10-volt 
battery,  what  resistance  must  the  coil  have?     (2  mm.) 

3.  A  rectangular  field  coil  10  X  16  inches  carries  a  current  of 
10  amperes  and  has  a  drop  of  40  volts.     The  surface  being  such 
that  s&tf  watt  is  given  off  per  sq.  cm.  and  degree  centigrade,  how 
long  must  the  coil  be,  not  to  have  a  rise  of  more  than  40  degrees 
above  the  temperature  of  the  air?     (2  mm.) 

4.  For  an  armature  of  a  certain  type  and  speed  there  are  0.02 
watt  radiated  per  sq.  in.  per  degree  F.  (area  taken  as  two  com- 
plete ends  and  the  cylindrical  surface).     The  diameter  is  10  inches 
and  the  length  8  inches.     The  resistance  is  0.04  ohm,  and  the 
hysteresis  and  eddy  loss  is  300.     What  is  the  temperature  rise  on  a 
full  load  of  100  amperes.     (2  mm.) 

5.  The  type  and  speed  of  a  10-pole,  550-kw.,  550-volt  generator 
(90  r.p.m.)  are  such  that  a  rise  of  temperature  of  85°  C.  in  the 
armature  would  dissipate  one  watt  per  sq.  cm.     If  the  available 
radiating  surface  of  the  armature  is  78,500  sq.  cm.,  the  resistance 
0.0125  and  the  iron  loss  11,000  watts,  required  the  temperature 
at  full  load. 

What  would  be  the  capacity  of  this  machine  if  rated  in  accord- 
ance with  the  A.I.E.E.  standardization  report?     (4  mm.) 

6.  A  6-pole,  150-kw.,  250-volt  generator  (same  machine  as  in 
problem  2,  Chap.  XII,  450  r.p.m.,  peripheral  velocity  2840  ft. 
per  minute)  has  an  armature  surface  of  such  a  character  that  it 
radiates  ^  of  a  watt  per  square  inch  and  per  degree  centigrade 


34      ELECTRICAL  ENGINEERING  PROBLEMS 

temperature  rise.  The  armature  resistance  is  0.008;  the  hystere- 
sis loss  in  the  teeth  is  620  watts  and  in  the  body  1140;  the  eddy 
current  loss  in  the  teeth  is  70  and  in  the  body  130.  The  radiating 
surface  is  2500  square  inches.  Required  the  temperature  rise. 

What  would  be  the  capacity  of  this  machine  if  rated  in  accord- 
ance with  the  A.I.E.E.  standardization  report?     (4  min.) 


CHAPTER  XIV 
DYNAMO   LOSSES   AND   EFFICIENCIES 

1.  A  10-kw.,  110-volt,  2-pole  shunt  generator  has  the  armature 
resistance  0.04  and  the  shunt  field  22.     Required  the  copper  loss 
at  full  load.     The  field  rheostat  resistance  is  15  ohms.     If  the 
machine  gives  110  volts  at  no  load  with  this  resistance  all  in, 
required  the  no-load  copper  loss  under  these  conditions.     (4  min.) 

2.  A  500-kw.,  550-volt  generator  requires  2.10  volts  to  send 
200  amperes  through  its  armature  when  at  rest.     What  will  be 
the  armature  copper  loss  when  giving  its  full-load  current  of  910 
amperes  at  550  volts?     (8  min.) 

3.  A  500-volt,  4-pole,  35-kw.  generator  (500  r.p.m.)  has  resist- 
ances as  follows:    Armature,  0.3;  shunt  field,  200;  series,  0.15. 
Required  the  copper  loss  for  a  load  of  60  amperes.     (4  min.) 

4.  If  in  the  generator  of  problem  1  the  iron  losses  constitute 
2%,  and  the  friction  loss  3%  of  the  rated  output,  required  the 
efficiency  at  full  load.     Also  for  J  load,  assuming  the  speed  to 
remain  constant,  and  that  8  ohms  of  the  rheostat  are  in  the  field 
circuit.     (3  min.) 

5.  In  the  generator  of  problem  3  if  the  hysteresis  loss  be  140 
watts,  that  due  to  eddy  currents,  50  watts,  and  if  the  friction  loss 
be  7J%  of  the  rated  capacity,  required  the  efficiency  of  the  machine 
for- a  load  of  60  amperes.     (2  min.) 

6.  If  the  machine  of  problem  6,  Chap.  XIII,  has  a  brush-contact 
resistance  =  0.003,  series-field  resistance  =  0.0008  and  shunt-field 
resistance  =  60,  required  the  total  copper  loss  for  no  load  and  for 
full  load.     (4  min.) 

7.  If  in  problem  6  the  total  friction  loss  be  5  kilowatts,  required 
the  full-load  and  half-load  efficiency.     (6  min.) 

8.  In  the  generator  of  problem  6,  Chap.  XIII,  the  outside  diam- 
eter of  the  armature  disks  is  83.5  cm.,  the  inside  diameter  45.7  cm., 
the  depth  of  a  slot  4.13  cm.,  and  the  iron  length  of  the  core  22.8 
cm.     Also  the  mean  density  is  10,700.     Required  the  hysteretic 
constant  to  give  the  hysteresis  loss  as  stated  in  watts.     (10  min.) 

9.  The  thickness  of  the  disks  in  the  generator  of  problem  8  is 

35 


36 


ELECTRICAL  ENGINEERING  PROBLEMS 


20  mils.     Required  the  eddy  current  constant  to  give  the  stated 
loss  in  watts.     (The  thickness  is  to  be  kept  in  mils.)     (10  min.) 

10.  A  110-volt,  100-ampere  armature  has  a  resistance  of  0.025 
ohm.     Plot  a  curve  of  loss  in  the  conductors  with  current;   give 
the  equation  for  the  resulting  curve  and  its  name.     If  the  machine 
be  a  long  shunt  compound,  100-volt  generator,  with  the  series- 
field  resistance  0.01  and  the  shunt-field  resistance  25,  plot  the  field 
losses  and  the  total  copper  loss  with  current  in  the  outside  circuit 
as  abscissae.     (10  min.) 

11.  A   1000-kw.,    1000-volt  generator  has   1000  r.p.m.   rated 
speed.     The  armature  resistance  is  0.106  ohm.      When  run  as  a 
motor  with  normal  field  strength  it  takes  925  volts  on  the  armature 
to  give  normal  speed  and  the  armature  current  is  22.2  amperes. 
What  is  the  stray  power?     (4  min.) 

12.  In  the  generator  of  problem  11  the  other  resistances  are 
series  field,  0.002;    interpole,  0.0035;    the  shunt  current  is  2.24 
amperes  and  the  brush  drop  (both  sides)  is  2.8.     Required  the 
efficiencies  for  J,  J,  f ,  1  and  1J  load.     Plot  these  as  a  curve  with 
output  as  abscissas.     (15  min.) 

13.  A  100-volt  shunt  motor  has  the  resistance  of  the  armature 
0.03  and  of  the  shunt  20  ohms.     The  full  load  on  the  armature  is 
10-kw.  input.     When  the  motor  is  run  on  no  load  at  full-load  speed, 
the  armature  takes  4  amperes  at  90  volts.     Required  the  full-load 
efficiency.     Neglecting  the  change  in  stray  losses,  find  also  the 
f ,  J,  \  and  TV  load  efficiencies  and  plot  with  input.     (15  min.) 

14.  A  test  of  a  2J-h.p.,  110-volt  shunt  motor  gives  the  following 
resistances:  Armature  0.3,  field  55. 

The  results  of  the  speed  load  test  are: 

ARMATURE 


Speed. 

Amp.  Input. 

Speed. 

Amp.  Input. 

1955 
1985 

21 
20 

2010 
2030 

19 

18 

The  results  of  the  iron-  and  friction-loss  test  are: 


Speed. 

Armature  current. 

Armature  volts. 

1950 
1975 
2000 
2025 

1.59 
1.65 
1.72 
1.80 

95 
98 
103 
108 

DYNAMO  LOSSES  AND  EFFICIENCIES  37 

Plot  a  curve  of  efficiency  and  output  and  find  from  it  the  effi- 
ciency for  2.2-h.p.  output.     (15  raw.) 

15.  A  120-volt  shunt  motor  is  giving,  as  determined  by  a  brake, 
an  output  of  23  h.p.     It  is  then  absorbing  20.2  kw.;    the  field 
current  is  5.3  amperes  and  the  armature  resistance  is  0.031  ohm. 
What  is  the  efficiency  and  power  intake  when  the  motor  is  giving 
10  h.p.?     (10  min.) 

16.  The  following  data  is  given  with  regard  to  the  armature 
of  the  500-kw.  generator  of  problem  2. 

Volume  of  iron  in  body  of  core,  cu.  cm 97,800 

Volume  of  iron  in  teeth .  26,050 

Density  in  body  of  core 17,000 

Density  of  teeth,  mean 21,000 

Speed,  r.p.m 330 

Poles 8 

Hysteresis  constant,  for  watts 2  X  10~10 

Eddy  loss 9350 

Determine  the  total  iron  loss  for  the  armature.     (8  min.) 

17.  The  shunt  field  of  the  500-kw.  generator  of  problem  16 
has  a  hot  resistance  of  89  ohms,  the  series  field  of  0.0022  and  the 
interpole  winding  of  0.0012,  the  brush  drop  at  full  load  is  1.6  volts, 
and  the  brush  friction  2280  watts.     The  bearing  and  windage 
friction  loss  is  3500  watts.     Required  the  one-half  load  and  full- 
load  efficiencies.     (10  min.) 

18.  A    105-volt,    4-pole    dynamo    gives    the    following    data: 
r.p.m.,  750;  gap  density,  13,000;  full-load  current,  300  amperes; 
length  of  armature  core,  50  cm.;    outside  diameter,  25  cm.;  and 
inside  diameter,  7.5  cm.     The  temperature  at  which  the  machine 
is  to  run  is  64°  C.     The  shunt  field  is  wound  with  2.25  kilometers 
of  No.  11  wire.     The  armature  resistance  is  0.005  ohm,  at  14°  C. 
The  drop  through  the  brushes  and  connections  at  full  load  is 
|  volt.     The  hysteresis  loss  in  the  armature  is  880  watts  per  cubic 
meter,  at  one  cycle  per  second.     The  friction  loss  is  400  watts. 
The  eddy  current  loss  is  2  watts  per  cubic  meter,  at  one  cycle  per 
second.      Required  the  iron  loss,  copper  loss,  total  loss  and  effi- 
ciency at  full  load.     (15  min.) 

19.  In  a  brake  test  upon  a  30-kw.,  500-volt  motor,  52  amperes 
are  supplied,  the  motor  is  making  1000  r.p.m.,  and  the  load  on  the 
3-foot  brake  arm  is  55  Ibs.     Required  the  efficiency  of  the  motor. 


38      ELECTRICAL  ENGINEERING  PROBLEMS 

What  change  in  the  length  of  the  brake  arm  could  be  made  to 
facilitate  the  computation  of  the  horse-power?     (2  min.) 

20.  A  rated  motor  is  being  used  as  a  dynamometer  to  measure 
the  power  absorbed  by  a  printing  press.     The  iron  losses  are  450 
watts,  the  field  resistance  493  ohms  and  the  armature  resistance 
0.53  ohm.      The  voltmeter  reads  512  volts  and  the  ammeter  in 
circuit  with  the  machine  21.3  amperes.     What  power  is  the  motor 
furnishing  to  the  press?     (5  min.) 

21.  The  motor  of  problem  13  is  to  be  used  to  drive  a  group  of 
tools  in  a  machine  shop.     All  the  tools,  including  a  punch  press, 
are  in  operation  one  hour  per  day  and  require  10  h.p.     Shutting 
down  the  press  reduces  the  load  to  5  h.p.,  at  which  it  runs  6  hours. 
During  the  remaining  2  hours  of  the  working  day  only  2  h.p.  are 
required,  and  for  an  hour  at  noon  only  1  h.p.  is  used.     Plot  a  curve 
of  total  loss  with  output  for  the  motor  and  hence  determine  its 
all-day  or  energy  efficiency.     (15  min.) 

22.  Two  550-volt,  1600-kw.  compound  dynamos  are  coupled 
together,  and  are  being  tested  by  the  Knapp  method,  that  is  the 
power  for  the   losses  is  supplied  electrically.     The  shunt-field 
currents  of  the  motor  and  generator  are  26.1  and  26.9  amperes. 
The  armature  resistance  of  each  machine  is  0.0037  ohm,  and  the 
average  of  the  armature  currents  is  the  rated  full-load  current. 
The  stray  losses  are  assumed  equal  in  the  two  machines.     322 
amperes  are  being  supplied  from  outside.     What  is  the  efficiency 
of  each  machine?     (10  min.) 


CHAPTER  XV 
MOTORS 

1.  A  wire  40  cm.  long  and  carrying  100  amperes  lies  in  a  field 
of  5000  gausses.     Required  the  force  acting  on  the  wire  in  dynes. 
(1  min.) 

2.  A  2-pole  motor  has  an  armature  diameter  of  8  inches  and 
an  armature  length  of  10  inches;   the  polar  angle  is  120  degrees 
and  the  width  of  the  pole  is  equal  to  the  armature  length;   there 
are  220  conductors  and  the  density  in  the  gap  is  4500.     Required 
the  horse-power  converted  by  the  motor  at  1200  r.p.m.,  and  when 
taking   40   amperes   in   the   armature.     (Neglect   the   fringing.) 
(5  min.} 

3.  The  following  data  refer  to  a  4-pole,  4-path,  220-volt  motor 
taking  140  amperes  in  its  armature  and  running  at  750  r.p.m.: 

Armature  diameter 15  inches 

Armature  length 11  inches 

Armature  conductors 300 

Polar  angle 72  degrees 

Gap  density 7500 

Required  the  horse-power  converted  and  also  the  horse-power 
output,  if  the  iron  losses  constitute  3%  and  the  friction  loss  2% 
of  the  output.  (7  min.) 

4.  What  horse-power  is  being  converted  by  a  motor  having 
124  conductors  on  a  6"  X  6"  armature,  the  poles  covering  120 
degrees  each,  and  the  total  flux  being  1.94  X  106  when  the  arma- 
ture current  is  22  amperes,  and  the  speed  is  1500  r.p.m.     (3  min.) 

5.  A  shunt  motor  has  220  volts  applied  and  takes  52.2  amperes. 
The  field  resistance  is  110  ohms  and  the  armature  resistance  is 
0.2  ohm.     Required  the  counter  e.m.f.     Also  if  the  total  flux  is 
2.72  X  106,  and  the  number  of  conductors  300,  required  the  number 
of  revolutions  per  minute.     (3  min.) 

6.  A  shunt  motor  on  a  110-volt  circuit,  having  an  armature 
resistance  of  0.25  ohm,  runs  on  no  load  at  1150  r.p.m.,  and  at  a 
full  load  of  30  amperes  at  1120.     With  30  amperes  how  much 

39 


40  ELECTRICAL  ENGINEERING  PROBLEMS 

resistance  would  have  to  be  put  in  series  with  the  armature  to 
reduce  the  speed  to  565  r.p.m.?  If,  with  this  resistance  in  circuit, 
the  current  be  next  reduced  one-half,  what  will  the  speed  become? 
(7  min.) 

7.  A  2-pole,  110-volt  motor  with  a  full-load  armature  current 
of  90  amperes,  has  an  armature  resistance  of  0.08  and  300  con- 
ductors;  and  the  total  flux  is  3.4  X  106.      Required  the  no-load 
speed,  the  full-load  speed,  and  the  per  cent  that  the  drop  in  speed 
is  of  the  full-load  speed  (the  regulation) .     The  brushes  are  midway 
between  the  poles,  and  the  no-load  current  is  3  amperes.    (4  min.) 

8.  In  problem  7  if  the  armature  resistance  were  doubled,  what 
would  be  the  speeds  and  the  per  cent  regulation?     Also  what  would 
be  the  speeds  and  regulation  if  by  reducing  the  field  current  the 
flux  were  reduced  to  3  X  106?     (5  min.) 

9.  If  in  problem  7  a  resistance  of  0.2  ohm  be  placed  in  the 
armature  circuit  for  the  purpose  of  reducing  the  speed,  what  will 
the  no-load  and  the  full-load  speeds  become?     What  per  cent  of 
the  power  used  will  be  consumed  in  the  resistance  at  full  load? 
(5  min.} 

10.  A  4-pole,  2-path,  220-volt,  24-h.p.  motor  has  a  full-load 
current  of  95  amperes.     The  field  current  is  5  amperes,  the  arma- 
ture resistance  is  0.03,  the  flux  is  4  X  106  and  the  conductors,  160. 
The  brushes  are  midway  between  the  pole  pieces  and  the  no-load 
current  is  4  amperes.     Required  the  no-load  and  full-load  speeds, 
and  the  per  cent  regulation.     (5  min.) 

11.  What  will  the  regulation  in  problem   10  become  if  the 
brushes  are  drawn  back  so  as  to  produce  back  ampere-turns  which 
reduce  the  flux  2%?     What  would  the  flux  have  to  be  made  in 
order  to  reduce  the  full-load  speed  10%?     (2  min.) 

12.  How  much  resistance  would  have  to  be  put  in  the  armature 
circuit  of  problem  10  in  order  to  reduce  the  speed  one  half?     What 
would  the  regulation  from  no  load  to  full  load  become  then? 
What  per  cent  of  the  power  supplied  would  be  lost  in  the  resist- 
ance?    (6  min.) 

13.  In  problem  10  how  much  starting  resistance  would  have 
to  be  put  in  series  with  the  armature  if  the  starting  current  were 
not  to  exceed  the  full-load  current?     If  the  motor  were  carrying 
half  load  current,  what  speed  would  it  reach  with  this  resistance 
in  the  circuit?     How  much  resistance  could  be  cut  out,  allowing 
the  current  to  again  reach  full-load  value?    (7  min.) 


MOTORS  41 

14.  A  4-pole,  220-volt,  25-h.p.  shunt  motor  is  wound  with  a 
2-circuit  winding  of  344  conductors  hi  43  slots,  and  runs  light  at 
800  r.p.m.     The  dimensions  of  the  conductors  are  90  X  200  mils. 
It  is  desired  to  use  this  frame  for  a  400-r.p.m.  110- volt  motor. 
Specify  two  windings  which  will  accomplish  this.     The  field  is 
wound  with  2200  turns  per  spool  of  wire  46  mils  in  diameter. 
Specify  the  changed  winding  necessary.     (4  min.) 

15.  The  resistance  of  the  armature  (unchanged)  of  problem  14 
is  0.067  ohm.     The  efficiency  of  the  motor  being  91%  and  the 
field  current  1.7  amperes,  what  will  be  the  speed  with  rated  load? 
(7  min.) 

16.  The  magnetization  curve  for  a  440-volt,  10-h.p.,  4-pole, 
800-r.p.m.  (at  no  load)  shunt  motor,  is  given  by  the  following  data: 

Ampere-turns 1000    2000    3000     4000 

Maxwells  ^-  106 0.53     0.97      1.24      1.48 

The  normal  ampere-turns  are  2500  for  440  volts.  At  what  speed 
will  the  motor  run  with  no  load  on  a  330-volt  circuit,  and  also  on  a 
550-volt  circuit?  Neglect  changes  in  the  resistance  of  the  shunt 
winding.  (7  min.) 

17.  The  normal  resistance  of  the  shunt  motor  in  problem  16  at 
60°  C.  is  25  ohms.     At  what  speed  will  the  motor  run  when  started 
up  out  of  doors  at  -  20°  C.?     (4  min.) 

18.  The  motor  of  problem  16  has  its  field  rewound  with  wire 
of  double  the  area  so  as  to  make  it  available  for  variation  of  speed 
by  the  field  rheostat.     What  must  be  the  resistances  hi  the  rheo- 
stat when  the  motor  is  running  at  600  and  1000  r.p.m.?     (4  min.) 

19.  If  in  problem  16  the  armature  resistance  is  1.4  ohms,  at 
what  speed  will  the  motor  run  with  rated  load  on  the  440-volt 
circuit?     (3  min.) 

20.  A  6-pole,  500-volt  shunt  motor  with  a  6-path  armature  has 
a  full  load  of  300  amperes.     The  armature  resistance  is  0.032 
ohm,  and  the  total  flux  at  no  load  is  8.96  X  106.     The  armature 
has  978  conductors,  and  the  field  excitation  is  15,000  ampere-turns 
per  pair  of  poles.     Assuming  the  brushes  to  be  midway  between 
the  poles,  required  the  speed  in  r.p.m.  for  currents  of  5  and  300 
amperes  in  the  armature.     (4  min.) 

21.  A   440-volt,    850-h.p.,    100-r.p.m.,    14-pole   motor   has   a 
parallel-wound  armature,  a  full-load  current  of  1550  amperes,  and 


42  ELECTRICAL  ENGINEERING  PROBLEMS 

an  active  conductor  length  of  33  cm.  The  mean  density  in  the 
slots  is  3600.  With  how  many  pounds  does  the  conductor  press 
against  the  side  of  the  slot?  If  the  area  of  the  air  gap  is  1360 
sq.  cm.  and  the  total  flux  is  15.5  megamaxwells,  what  would  be 
the  pull  on  the  conductor  if  the  armature  were  smooth  core? 
(4  min.) 

22.  A  220-volt  shunt  motor  is  to  be  used  as  a  dynamometer  to 
measure  the  power  taken  by  a  lathe  making  a  heavy  cut.     The 
speed  is  1060,  the  current  is  23.2,  and  the  voltage  on  the  armature 
is  218.     With  218  volts  on  the  field  and  209  on  the  armature  it  runs 
without  load  at  1060  r.p.m.  and  takes  1.7  amperes.    The  resistance 
of  the  armature  is  0.62  ohm.     Required  the  power  taken  by  the 
lathe.     (5  min.} 

23.  Two  10-h.p.,  1200-r.p.m.,  220-volt  motors  are  mechanically 
coupled  and  have  their  armatures  connected  in  series  on  a  220-volt 
circuit.     The  fields  are  each  normally  excited.     At  what  speed  will 
the  motors  run,  and  how  many  horse-power  will  the  combination 
produce?     (2  min.} 

24.  Same  as  problem  23,  except  that  one  motor  is  a  220-volt 
and  the  other  a  110-volt  motor.     (3  min.) 

25.  Same  as  problem  23,  except  that  the  field  strength  of  one 
motor  is  increased  25%.     (4  min.) 

26.  A  4-pole,  2-path,   500-volt  series   motor   has   a  full-load 
rating  of  20  amperes.     The  armature  and  field  resistances  are 
1  and  1.4  ohms  and  there  are  920  conductors.     The  field  turns  per 
pair  of  poles  are  300.     Its  magnetization  curve  is  as  follows: 

Ampere-turns .  .  .  3000    4000    5000    6000    7000    8000    9000 
Megamaxwells  ..  1.76    2.27    2.72    3.02    3.14    3.25    3.34 

Required  the  speed  for  half  load,  full  load  and  50%  over  load. 
The  brushes  are  midway  between  the  pole  tips.     (6  min.) 

27.  If  a  resistance  of  10  ohms  is  put  in  series  with  the  motor 
of  problem  26,  at  what  speeds  will  it  run  with  full-load  current  and 
with  half-load  current?     (3  min.) 

28.  If  a  shunt  of  1.4  ohms  is  put  around  the  field  in  problem  26, 
at  what  speed  will  it  run  when  taking  20  amperes  in  the  armature? 
(2  min.) 

29.  Given  a  4-pole,  500-volt,  35-h.p.  railway   motor  with   a 
2-path  winding,  44  field  turns  per  pole,  744  armature  conductors, 
armature  resistance  0.32  and  field  resistance  0.168.     The  brushes 


MOTORS  43 

are  set  midway  between  the  pole  tips.     The  magnetization  curve 
is  as  follows: 

Flux  per  pole 1X106    2X106    3X106    3.5X106    4X106 

Ni,  per  2  poles 1160       2180       3400        4200         5250 

Find  the  speeds  for  2,  30  and  60  amperes.     (10  min.) 


PART   II 

ALTERNATING-CURRENT  CIRCUITS 
AND  APPARATUS 


PART  II 

CHAPTER  I 
INDUCTANCE   AND    INDUCED    E.M.F. 

1.  A  current  of  25  amperes  changes  in  T\j  of  a  second  to  5 
amperes;    required  the  rate  of  change  of  the  current  during  this 
time,     (i  min.) 

2.  When  five  amperes  flow  through  a  circuit  surrounding  an 
iron' ring,  there  is  a  flux  of  105  maxwells;    when  the  current  is 
increased  to  10  amperes,  the  flux  increases  to  1.6  X  105.     Required 
the  average  rate  of  change  of  flux  with  current  within  the  above 
current  limits.     (%  min.) 

3.  During  the  first  10  minutes  a  man  goes  1J  miles,  during  the 
next  20  minutes  he  goes  2  miles  and  in  the  next  half  hour  he  goes 
1^  miles.     What  are  his  average  speeds,  or  rates  of  change  of  dis- 
tance with  time,  during  the  first  30  minutes  and  during  the  whole 
time?    Assuming  a  regular  decreasing  speed,  what  is  his  instan- 
taneous speed  at  the  end  of  20  minutes?     (1  min.) 

4.  At  a  certain  point  in  a  city  property  sells  for  $120  per  front 
foot.     Half  a  mile  farther  out  its  value  has  dropped  to  $80,  at  1  mile 
it  is  $60,  at  2  miles  it  is  $30  and  at  3  miles  $20.     Find  the  average 
rate  of  change  of  value  with  distance  within  the  above  limits. 
Find  also  the  rate  (instantaneous)  at  which  values  are  changing 
at  the  2-mile  point.     (5  min.) 

6.  A  ring  is  built  of  wrought-iron  rod,  having  an  area  of  20 
sq.  cm.  The  mean  diameter  is  25  cm.  and  there  are  2000  turns 
in  the  winding.  Required  the  average  rates  of  change  of  flux 
density  and  total  flux,  with  current,  when  the  exciting  current  is 
brought  from  zero  to  values  of  0.25, 1,  3  and  5  amperes  respectively. 
Use  the  curve  for  wrought  iron  given  on  page  105.  (15  min.) 

6.  In  the  case  of  problem  5,  find  the  instantaneous  rate  of 
change  of  flux,  when  the  current  is  passing  through  a  value  of  one 
ampere.     Also   find   the   average   rate   between   one   and   three 
amperes.     (5  min.) 

7.  A  conductor  in  the  form  of  a  loop  is  threaded  by  106  lines  of 

47 


48  ELECTRICAL  ENGINEERING  PROBLEMS 

force;   the  current  is  broken  in  Ti<y  second;   required  the  average 
induced  electromotive  force  set  up,  in  volts.     (J  min.) 

8.  A  coil  of  200  turns  of  wire  is  threaded  by  100,000  lines  of 
force.     The  exciting  current  is  halved  in  T£¥  second;  required  the 
induced  e.m.f.     If  the  original  exciting  current  is  5  amperes,  what 
is  the  inductance  of  the  circuit?     (2  min.} 

9.  The  inductance  of  a  certain  dynamo  is  40  henries;  what  will 
be  the  induced  e.m.f.  if  the  exciting  current  of  2  amperes  is  brought 
to  zero  in  0.025  second?     (1  min.) 

10.  What  is  the  average  rate  of  change  of  current  per  second 
during  a  quarter  of  a  cycle  of  a  60-period  alternating  current  of 
100  amperes  (effective)?     (1  min.) 

11.  If  in  problem  5  each  one  of  the  changes  of  current  men- 
tioned takes  place  in  T^  second,  what  will  be  the  e.m.f.  set  up  in 
each  case?     (3  min.) 

12.  If  in  problem  6  the  current  is  changing  at  the  rate  of  50 
amperes  per  second  when  it  passes  through  one  ampere,  what 
e.m.f.  will  be  induced  at  this  instant?     (1  min.) 

13.  With  an  exciting  current  of  7.5  amperes  through  1000  turns 
the  reluctance  of  a  magnetic  circuit  is  0.1257.     If  the  current 
increases  by  2%  of  this  value,  the  reluctance  increases  1%.     If 
this  increase  of  current  is  at  the  rate  of  15,000  amperes  per  second, 
required  the   e.m.f.   induced.     What  is  the  average  inductance 
during  this  increase?     (3  min.) 

14.  The  core  of  a  25-kv-a.,  60-period  transformer  is  15  X  20 
cm.  and  the  primary  winding  has  500  turns.     The  density,  corre- 
sponding to  a  magnetizing  current  of  0.2  ampere,  is  5000.     Re- 
quired the  average  inductance  for  this  current,  and  the  induced 
e.m.f.  if  the  current  were  reduced  to  zero  at  a  rate  of  100  amperes 
per  second.     (3  min.) 

15.  An  electromagnet  has  1200  turns;    with  4.5  amperes  the 
flux  is  5  X  107,  and  with  2  amperes  3  X  107.     What  is  the  average 
inductance?     If  the  density  is  such  that  between  these  limits  the 
magnetization  curve  can  be  taken  as  a  straight  line,  what  will  be 
the  e.m.f.  of  inductance  if  the  current  change  from  3  to  3.2  amperes 
in  0.1  second?     (3  min.) 

16.  A  circuit  has  1000  turns,  the  area  of  the  enclosed  magnetic 
circuit  is  20  sq.  cm.     With  4  amperes  the  density  is  10,000  and 
with  9  amperes  it  is  14,000.     Required  the  average  value  of  L 
between  4  and  9  amperes.     (1  min.) 


INDUCTANCE  AND  INDUCED  E.M.F.  49 

17.  An  air  magnetic  circuit  has  753  turns  and  is  threaded  by 
646,000  maxwells  when  the  exciting  current  is  8.4  amperes.     If  in 
0.23  second  the  current  is  increased  from  this  value  to  97.3  am- 
peres, what  inductive  e.m.f.  will  result?     (4  min.) 

18.  A  dynamo  field  has  7320  turns,  is  9|  inches  in  diameter 
and  has  a  'density  of  16  kilogausses.     If  the  exciting  current  is 
broken  in  0.13  second  and  the  density  fall  to  a  residual  of  900 
gausses,  to  what  value  will  the  e.m.f.  at  the  terminals  of  the  field 
circuit  rise?     Also  if  2.6  seconds  be  allowed  for  the  breaking  of 
the  field  circuit?     (6  min.) 

19.  A  wrought-iron  ring,  with  2340  turns,  an  area  of  0.98  sq. 
cm.  and  a  cut  (air  gap)  across  it  of  J  inch,  has  14.0  kilogausses  for 
a  current  of  4.72  amperes,  what  is  the  average  inductance?    (3  min.) 

20.  In  problem  19  what  would  be  the  inductance  if  the  cut 
were  increased  to  J  inch  by  forcing  the  ends  apart?     Also  if  the 
ends  were  forced  together  so  as  to  make  the  air  gap  negligible? 
(6  min.) 

21.  A  wrought-iron  ring,  1.13  feet  in  inside  diameter  and  0.872 
inch  diameter  of  iron,  has  one  layer  of  No.  30  double-cotton-covered 
magnet  wire.     If  current  enough  is  used  to  give  a  density  of  8000 
gausses,  what  will  be  the  number  of  turns,  the  current  and  the 
average  inductance?     See  table  of  magnet  wire  on  page   104. 
(5  min.) 

22.  If  in  problem  21  the  density  is  doubled,  what  will  the 
current  be  and  what  will  the  inductance  become?     (2  min.) 

23.  A  ring  of  cast  iron  20  inches  in  mean  diameter  and  made  of 
2-inch  round  iron  is  wound  with  3250  turns.     Find  the  mean  value 
of  L  for  0.2,  0.5,  1  and  2  amperes  and  plot  the  curve  of  inductance 
with  current.     (15  min.) 

24.  Required  the  inductance  of  a  winding  of  one  layer  of 
No.  18  double-cotton-covered  magnet  wire  upon  a  wooden  ring, 
7.5  inches  mean  diameter  and  0.763  inch  diameter  of  cross-section. 
If  the  resistance  be  0.375  ohm,  what  is  the  time  constant?     (6 
min.) 

25.  A  certain  100-kv-a.  transformer  for  5000  to  500-volt  trans- 
formation has  a  maximum  flux  of  2.0  X  106  with  a  cross-section  of 
300  sq.  cm.     The  secondary  turns  are  110  and  the  primary  1100, 
the   effective  exciting   current   is  0.333   ampere.     Required  the 
average  inductance  of  each  winding  between  maximum  flux  and 
zero  flux.     (1  min.) 


50  ELECTRICAL  ENGINEERING  PROBLEMS 

26.  The  shunt  field  of  a  110-volt  dynamo  has  a  resistance  of 
42  ohms  and  an  inductance  of  12  henries.     If  100  volts  be  applied 
to  the  terminals,  how  long  will  it  take  the  current  to  reach  0.95 
of  its  maximum  value?     How  long  to  reach  0.99?     (4  ram.) 

27.  A  circuit  has  an  inductance  of  100  henries.     Plot  curves  of 
current  rise  with  an  applied  constant  e.m.f.  of  100  volts  for  the 
following  resistances,  1,  10,  100  and  1000  ohms.     Use  different 
scales  so  as  to  bring  the  final  current  value  the  same  in  each  case. 
(30  min.} 

28.  If  the  exciting  current  at  110  volts  in  problem  26  be  broken 
in  ^V  °f  a  second  and  the  current  thereby  reduced  to  zero,  what 
e.m.f.  will  exist  between  the  terminals?     (1  min.} 

29.  Two  eoite,  A  and  B,  lie  in  pastel  planes;  60%  of  th^lmes 
produced  by  one  coil  thread  through  the  other.     Five  amperes 
through  the  coil  A  produces  a  flux  of  5000  in  A.     If  the  current 
in  A  change  from  +6  to  —6  amperes  in  0.01  second,  what  will  be 
the  e.m.f.  produced  in  coil  B,  if  B  have  1000  turns?     What  is  the 
mutual  inductance?     (3  min.) 


CHAPTER  II 
QUANTITY   AND    CAPACITY,    CONDENSERS 

1.  A  current  of  5  amperes  flows  for  one-half  hour;   how  many 
coulombs  are  passed?     To  what  e.m.f .  would  this  raise  a  20-micro- 
farad  condenser?     A  50-microfarad  condenser  is  charged  to  2000 
volts;   how  long  would  this  condenser  maintain  a  current  of  one 
ampere  if  closed  through  an  outside  circuit?     (2  raw.) 

2.  An  electric  motor  takes  1  kilowatt  at  200  volts.     How  many 
coulombs  will  it  use  in  2  hours?     How  many  if  the  voltage  be  500 
and  the  power  the  same?     (2  ram.) 

3.  How  many  coulombs  and  how  many  watt-hours  are  repre- 
sented by  the  charge  in  a  100-microfarad  condenser  on  a  10,000- 
volt  circuit?     What  will  each  of  these  become  if  the  voltage  is 
doubled?     (4  raw.) 

4.  To  what  voltage  will  5  coulombs  charge  a  1-microfarad  con- 
denser?    A  100-microfarad  condenser?     (1  raw.) 

5.  How  long  would  it  take  to  charge  a  10-microfarad  condenser 
on  a  50,000-volt  circuit,  if  the  average  current  were  2  amperes? 
How  long  to  charge  with  the  same  current  a  100-microfarad  con- 
denser on  a  100-volt  circuit?     (2  raw.) 

6.  What  would  be  the  capacity  in  microfarads  of  a  condenser 
which  would  require  2000  volts  to  bring  to  a  charge  of  1  coulomb? 
(J  raw.) 

7.  What  capacity  would  be  necessary  to  run  a  25-watt  tungsten 
lamp  for  5  minutes  at  100  volts?     If  a  2-microfarad  condenser 
costs  90  cents  what  would  the  above  cost?     (3  raw.) 

8.  What  voltage  would  be  required  to  charge  a  50-microfarad 
condenser  with  one  coulomb  of  electricity?     (1  raw.) 

9.  In  0.02  second  the  charge  in  a  condenser  changes  from  0.283 
to  0.231  coulomb;  required  the  average  current  flowing.     (J  raw.) 

10.  A  10-microfarad  condenser  shows  a  drop  in  voltage  from 
3000  to  1500  in  0.03  second;  required  the  average  current.    (1  raw.) 

11.  Three  condensers  each  having  a  capacity  of  30  microfarads 
are  in  parallel.     What  is  the  combined  capacity?     What  would 
be  the  capacity  if  they  were  in  series?     (J  raw.) 

51 


52  ELECTRICAL  ENGINEERING  PROBLEMS 

12.  Two  condensers  of  10  and  15  microfarads  are  in  parallel; 
what  is  the  combined  capacity?     What  would  it  be  if  they  were 
in  series?     (1  ram.) 

13.  Two  condensers  have  capacities  of  1  and  20  microfarads; 
what  are  the  combined  capacities  when  they  are  in  parallel,  and 
when  they  are  in  series?     (1  min.) 

14.  With  the  condensers  of  problem  13  in  series  on  a  1000-volt 
.  circuit,  what  would  be  the  voltage  around  each  condenser?     How 

could  this  arrangement  be  used  in  voltage  measurement?     (3  min.) 

15.  A  50-microfarad  condenser  charged  to  -2000  volts  is  con- 
nected to  a  non-inductive  circuit  of  100  ohms.     Obtain,  by  approxi- 
mate method,  and  plot  a  curve  between  terminal  e.m.f.  and  time. 
Get  points  for  1,  2,  3,  4,  6  and  10  thousandths  of  a  second;  make 
approximations  enough  to  locate  the  curves  within  2%.     (10  min.) 

16.  From  the  curves  obtained  in  problem  15  determine  the 
instantaneous  value  of  the  current  at  0.0024  second  after  the 
beginning  of  the  discharge.     (1  min.) 

17.  The  condenser  of  problem  15  is  connected  through  the  100- 
ohm  circuit  to  a  2000-volt  e.m.f.     Obtain  the  charge  at  the  end  of 
1,  2,  3,  4,  6  and  10  thousandths  of  a  second  and  plot  the  same. 
(12  min.) 

KA 

18.  Using  the  formula:    Capacity  =  ^ — ,  5  microfarad, 

OU  7TU.    X    1" 

where  A  and  d  are  area  and  distance  between  plates  in  centimeters 
and  K  is  the  inductivity  of  the  dielectric,  determine  how  many 
square  meters  of  glass  plates  3  mm.  thick  with  copper  foil  on 
each  side  would  have  to  be  used  to  provide  a  0.045-microfarad 
condenser  for  a  5-kw.,  25,000-volt  wireless  sending  outfit.  Take 
the  inductivity  of  the  glass  used  as  6.  Neglect  the  capacity 
between  the  adjacent  plates.  (3  min.) 

19.  As  an  alternative  form  of  condenser  for  the  above  case, 
determine  the  number  of  metal  plates  40  centimeters  square,  that 
would  have  to  be  immersed  in  petroleum  with  an  inductivity  of  2.2 
and  spaced  1  centimeter  apart  to  give  the  same  capacity.    (4  min.) 


CHAPTER  III 
ALTERNATORS   AND    WAVE  FORMS 

1.  Required  the  frequency  in  periods  per  second  of  a  current 
given  by  an  8-pole  alternator  running  900  r.p.m. ;  also  of  a  28-pole 
machine  running  180  r.p.m.     (1  min.) 

2.  A  generator  is  to  be  driven  by  an  engine  running  120  r.p.m.; 
how  many  poles  must  it  have  to  give  a  frequency  of  60?     A  40- 
period  generator  has  16  poles;  at  what  speed  does  it  run?     (1  min.) 

3.  What  is  the  speed  of  a  two-pole  turbo-generator  for  a  60- 
period  system?     Also  for  a  25-period  system?     (1  min.) 

4.  Find  a  number  which  when  divided  by  the  number  of  poles 
of  any  60-cycle  generator  or  motor  will  give  the  synchronous  speed 
in  r.p.m.     Also  for  a  25-cycle  machine.     (1  min.) 

5.  Construct  a  table  giving  the  speeds  for  machines  from  2  to 
36  poles  for  60  and  for  25  cycles.     (10  min.) 

6.  What  are  the  kilowatt  capacities  of  a  100-kv-a.  generator 
with  currents  lagging  30,  45  and  60  degrees?     (1  min.) 

7.  An  engine  builder  whose  engine  runs  at  650  r.p.m.  demands 
a  60-cycle  alternator  for  direct  coupling.     What  can  be  done  for 
him?     (2  min.) 

8.  By  the  rotation  of  a  vector  and  taking  its  projections,  plot 
the  sine  wave  of  current,  i  =  100  sin  a.     (5  min.) 

9.  By  division  into  narrow  strips  and  measuring  the  mean 
ordinates  obtain  the  average  ordinate  for  a  half  cycle.     (10  min.) 

10.  Construct  the  curves  of  squared  ordinates  and  by  measure- 
ment of  the  surface  obtain  the  mean  ordinate,  and  hence  the  square 
root  of  the  mean  square  ordinate.     (15  min.) 

11.  Construct  the  following  curves  and  explain  which  might 
be  the  wave  form  of  an  alternator  and  why: 

e  =  100  sin  a  +  20  sin  2  a, 

e  =  100  sin'a  +  20  sin  (3  a  -  30°).  (40  min.) 

12.  Find  the  effective  value  of  a  wave  made  up  of  straight  lines 
rising  from  zero  to  a  100-volt  maximum  and  falling  to  zero  again, 

53 


54  ELECTRICAL  ENGINEERING  PROBLEMS 

and  obtain  the  ratio  of  this  to  the  average  value.     Compare  this 
with  the  same  ratio  for  the  sine  wave.     (10  min.) 

13.  Find  the  effective  value  of  the  wave  form  e  =  100  sin  a  + 
20  sin  (3  a  —  30°)  as  constructed  in  problem  11,  and  obtain  its 
ratio  to  the  average  value  of  the  same  curve.  (15  min.) 


CHAPTER  IV 

ALTERNATING   CURRENT   IN   INDUCTIVE   CIRCUITS 

1.  In  a  given  circuit  the  resistance  is  10,  the  inductance  0.01 
and  the  frequency  60.     Required  the  e.m.f.  necessary  to  cause  2 
amperes  to  flow.     What  current  would  flow  if  a  direct-current 
e.m.f.  of  the  same  value  were  applied  to  the  circuit?     (3  ram.) 

2.  In  a  certain  circuit,  r  =  1,  L  =  0.01  andco  =  400  (frequency 
=  63.7);  required  the  current  and  angle  of  lag  if  50  volts  are 
applied.     What  would  be  the  current  if  this  e.m.f.  were  direct 
current?     Give  the  equation  for  the  instantaneous  value  of  the 
current.     (4  ram.) 

3.  In  a  certain  circuit  r  is  2  and  L  is  0.02  and  a  100-volt  alter- 
nating e.m.f.  gives  10  amperes;    required  the  frequency  of  this 
current;  also  the  reactance  and  impedance  of  the  circuit.     (3  ram.) 

4.  In  a  certain  circuit  r  is  2  ohms,  and  with  a  200-volt  50-period 
circuit  50  amperes  flow;    required  the  values  of  the  inductance, 
reactance  and  impedance.     (3  ram.) 

5.  The  exciting  current  of  a  100-volt,  3J-kw.  shunt  generator 
field  is  2  amperes,  and  its  inductance  is  10  henries;  required  what 
this  current  would  be  if  an  equal  60-period  e.m.f.  were  applied  to 
it.     Also  if  the  alternating  current  were  of  120  periods.     (7  ram.) 

6.  The  primary  circuit  of  a  2000-volt,  2-kw.  transformer  has 
a  resistance  of  20  ohms,  and  an  inductance  of  40  henries.     What 
current  will  flow  through  this  coil  on  a  60-period,  2000-volt  circuit, 
the  secondary  circuit  being  open,  and  what  will  be  the  angle  of  lag. 
What  current  would  flow  with  a  direct  current  of  equal  e.m.f.? 
Give  the  equation  for  instantaneous  current  values.     (6  ram.) 

7.  A  10-ohm  coil  is  placed  upon  a  100-volt  circuit  in  which  co  is 
400;    it  is  found  that  the  current  lags  30  degrees;    required  the 
current,  the  inductance  and  the  reactance  of  the  circuit.     (4  ram.) 

8.  In  the  circuit  of  problem  10  the  e.m.f.  is  raised  until  200 
amperes  flow,  what  will  it  become?     (2  ram.) 

9.  A  current  of  3000  amperes  delivered  by  a  100-volt  trans- 
former with  co  =  400,  has  a  power  factor  of  80%;    required  the 
resistance  and  reactance  of  the  circuit.     (3  ram.) 

55 


56  ELECTRICAL  ENGINEERING  PROBLEMS 

10.  If  a  milliammeter  with  inductance  in  series  be  used  as  a 
frequency  meter,  upon  a  circuit  of  constant  e.m.f .  of  120  volts,  and 
if  the  resistance  is  negligible  and  the  inductance  is  10,  what  current 
will  indicate  a  frequency  of  60  periods?     (2  min.) 

11.  Given  E,   127;    r,  40;    I,   1.54;  and  the  frequency,   115; 
required  the  reactance,  inductance  and  angle  of  lag.     (6  min.) 

12.  Given  E,  1100;    I,  6.54;    r,  86;  and  the  frequency,  115; 
required  the  reactance  and  inductance.     (6  min.) 

13.  Given  an  impressed  e.m.f.  of  100;  r,  5;  and  1, 12;  determine 
the  value  of  L  and  the  induced  e.m.f.  if   the  frequency   is  63. 
(3  min.) 

14.  Required  the  e.m.f.  to  send  10  amperes  through  a  circuit 
with  3  ohms  if  the  e.m.f.  of  inductance  is  20.     (1  min.) 

15.  Given  for  a  coil  of  wire  r,  40  and  L,  0.05;  if  the  frequency 
is  95.5,  required  the  angle  of  lag  as  denned  by  the  tangent  and 
the  e.m.f.  necessary  to  send  10  amperes  through.     Construct  the 
angle  of  lag.     (6  min.) 

16.  An  inductive  circuit  having  a  resistance  of  2  ohms  carries 
10  amperes,  maximum,  and  an  inductive  e.m.f.  of  10  volts  maxi- 
mum is  set  up  in  the  circuit.     Construct  the  sine  curves  for  active 
e.m.f.  and  induced  e.m.f.  and  from  these  derive  and  construct  the 
curve  of  impressed  e.m.f.     (10  min.) 

17.  A  circuit  of  5  ohms  resistance  carries  20  amperes  maximum; 
an  inductive  e.m.f.  of  30  volts  is  set  up  in  the  circuit.     Construct 
the  e.m.f.  parallelogram  for  this  case,  and  by  revolution  of  this 
construct  the  sine  curves  for  the  active,  induced  and  impressed 
e.m.f  .'s.     (10  min.) 

18.  Given  for  a  circuit  resistance  4  ohms,  maximum  current 
20  amperes,  maximum  inductive  e.m.f.  20  volts;   construct  e.m.f. 
curves,  including  the  impressed  e.m.f.,  and  from  these  the  parallelo- 
gram of  e.m.f.'s.     (10  min.) 

19.  Given  an  effective  current  of  20  amperes  in  a  circuit  whose 
resistance  is  4  and  reactance  2  ohms;  plot  the  curves  of  active  and 
induced  e.m.f.  and  from  them  the  curve  of  impressed  e.m.f.     Also 
construct  the  e.m.f.  parallelogram  which  would  generate  these 
curves.     (10  min.) 

20.  In  a  circuit  of  10  ohms  and  0.005  henry,  what  is  the  largest 
value  that  the  frequency  can  have,  without  producing  a  decrease 
of  more  than  1%  in  the  current  which  flows  when  the  frequency 
is  zero?     (5  min.) 


ALTERNATING  CURRENT  IN  INDUCTIVE  CIRCUITS     57 

21.  An  instrument  for  use  on  a  circuit  with  w,  400  (frequency, 
63.7)  has  r,  1000,  and  L,  1.     What  per  cent  increase  of  frequency 
would  cause  an  error  of  1%?     (7  min.) 

22.  Given  a  laminated  ring  30  cm.  in  circumference,  3  cm.  in 
diameter  of  cross-section  and  having  n  =  2000,  at   a  density  of 
B  =  6000;  required  the  number  of  turns  to  give  L  =  0.1.     (4min.) 

23.  Given  for  an  inductive  circuit  r,  5;  L,  0.01;   E,  75;  and  I, 
11.8;  required  the  frequency.     (4  min.) 

24.  A  solenoid  of  800  turns,  1  meter  long  and  5  cm.  in  diameter, 
has  r,  0.15  ohm;  required  the  inductance  and  the  current  when  a 
10-volt,  75-period  current  is  applied,  and  also  the  angle  of  lag. 
What  would  the  current  be  if  the  same  wire  were  wound  non- 
inductively?     If  the  frequency  be  doubled,  what  will  be  the  cur- 
rent and  the  angle  of  lag?     (15  min.) 

25.  Given  for  an  inductive  circuit  r,  40.4;  L,  0.101;  frequency, 
100;  and  I,  1.34;   required  the  impedance,  the  e.m.f.  and  the  angle 
of  lag.     (7  min.)  • 

26.  A  circuit  has  0.025  henry  and  a  variable  resistance.     115 
volts  having  a  periodicity  of  58  is  applied;   construct  the  locus  of 
the  current  vector,  and  show  graphically  that  it  is  correct  for  two 
cases.     (5  min.) 

27.  A  circuit  on  a  100-volt  system  has  a  resistance  of  5  ohms. 
Plot  the  locus  of  current  with  varying  reactance.     Note  that  the 
change  in  reactance  may  be  due  either  to  change  in  inductance  or 
frequency,  so  that  the  same  curve  could  be  used  for  either.     (5 
min.) 

28.  A  condenser  of  50  microfarads  capacity  is  connected  to 
a  60-period,  1 10-volt  circuit.     What  charging  current  will  flow? 
What  will  it  become  on  1100  volts.     What  will  be  the  ratio  of 
volt-amperes  in  the  two  cases?     (4  min.) 

29.  If  the  condenser  of  problem  28  is  used  on  a  1200-period, 
1 100-volt  circuit,  what  will  the  charging  current  be?     What  will 
be  the  capacity  reactance?     (3  min.) 

30.  A  25-period,  100,000-volt,  110-mile  transmission  line  has 
an  equivalent  capacity  of  0.01  microfarad  per  mile.     What  will 
be  the  charging  current  and  the  charging  kilovolt-amperes?     What 
is  the  capacity  reactance?     (3  min.) 

31.  A  100-microfarad  condenser  is  connected  to  an  alternating- 
current  generator  having  o>,   400,  through  a  line  of  negligible 
resistance;  the  maximum  current  is. 8.     Plot  sine  curves  of  current 


58  ELECTRICAL  ENGINEERING  PROBLEMS 

and  condenser  and  generator  e.m.f. 's  in  their  proper  phase  relation. 
Construct  a  vector  diagram  for  these  three  curves.     (10  min.) 

32.  The  e.m.f.  curve,  e  =  100  sin  a  +  20  sin  (3a  -  30°),  con- 
structed in  problem  11,  Chapter  III,  has  co  =  400  for  its  funda- 
mental, and  is  applied  to  a  circuit  having  an  inductance  of  0.05 
henry  and  a  resistance  of  2  ohms.     Plot  the  e.m.f.  and  the  current 
curve  in  their  proper  phase  relation.     (30  min.) 

33.  The  same  e.m.f.  as  in  problem  32  is  applied  to  a  100-micro- 
farad  condenser.     Plot  the  curves  of  e.m.f.  and  current  in  their 
proper  relation.     (25  min.) 


CHAPTER  V 
INDUCTIVE  .CIRCUITS  IN  SERIES  AND  PARALLEL 

(Solve  all  problems  graphically  and  give  results  in  numbers. 
Angles  may  be  defined  by  their  tangents.) 

1.  Two  circuits  having  resistances  of  2  and  10  ohms  respectively 
and  ^  and  T£^  henries  are  in  series  on  a  circuit  having  co,  300; 
find  the  e.m.f.  around  each  of  these  and  across  the  outside  ter- 
minals with  200  amperes  flowing.   What  is  the  frequency?    (5  min.) 

2.  An  induction  coil  is  to  be  used  for  cutting  down  the  light  of 
a  100- volt  lamp;  the  voltage  is  to  be  reduced  to  50,  with  which 
the  current  through  the  lamp  is  0.2  ampere.     If  the  coil  have  10 
ohms  resistance,  determine  what  will  be  the  e.m.f.  induced  in  it, 
and  what  will  be  its  terminal  e.m.f.     What  per  cent  of  the  energy 
used  will  be  consumed  in  the  coil?    What  is  the  power  factor? 
(4  min.) 

3.  Two  inductive  circuits  in  series  are  designated  by  ri,  r2,  Xi, 
x2,  Ei,  £2,  E  and  I.     In  general,  if  five  of  these  are  given  the  other 
three  can  be  determined  graphically  from  the  triangle  diagram. 
Draw  and  letter  this  diagram  and  then  state  briefly  but  completely 
the  geometrical  construction  for  the  cases  where  the  following  are 
the  unknown  quantities:    (a)  PI,  r2,  x2;    (b)  r2,  x2,  E;    (c)  r2,  X2,  I; 
(d)  x2,  Ei,  E,;   (e)  x2,  E,  I;   (f)  EI,  E2,  I.     (30  min.} 

4.  Given  the  resistances  and  reactances  of  three  circuits  in 
series,  and  the  line  e.m.f.     Show  how  to  determine  the  current  and 
the  e.m.f.  for  each  circuit.     (5  min.) 

6.  Given  two  circuits  having  FI,  5;  LI,  0.01;  r2, 20;  and  L2, 0.02; 
if  an  alternating  current  of  500  amperes  with  <o,  600,  be  put  through 
these  two  circuits  in  series,  what  e.m.f.  must  be  used  and  what  will 
be  the  drop  and  lag  for  each?  (8  min.) 

6.  Three  circuits,  A,  B  and  C,  are  in  series  and  have  respectively 
0.1,  0.01  and  0  henry  and  5,  15  and  12  ohms.     With  5  amperes 
at  60  periods  find  the  drop  around  A  and  B,  B  and  C,  and  A,  B 
and  C.     (7  min.) 

7.  Circuits  A,  B  and  C  in  series  have  inductances  of  1.2,  0  and 
0.3  and  resistances  of  0,  50  and  100;  also  a  100-period  current  gives 

59 


60  ELECTRICAL  ENGINEERING  PROBLEMS 

a  drop  of  1000  volts  around  A;  required  the  drops  around  B  and 
C  in  series  and  A,  B  and  C  in  series.     (7  min.) 

8.  Two  circuits  in  series  have  resistances  of  10  ohms  each,  one 
has  10  ohms  reactance  and  the  e.m.f.  across  the  other  is  15;  with 
1  ampere  flowing,  find  the  impressed  e.m.f.  and  the  reactance  of 
the  other  circuit.     (5  min.) 

9.  Two  circuits  in  series  have  reactances  of  2  and  4  ohms,  the 
first  has  3  ohms  resistance;   if  50  amperes  flow  when  the  circuits 
are  placed  on  a  400-volt  circuit,  required  the  value  of  the  other 
resistances.     (5  min.) 

10.  Given  two  circuits  in  series,  one  of  10  ohms  resistance  and 
the  other  of  0.1  henry  inductance;    required  the  current  if  E  be 
1000  and  «,  600.     Also  if  the  first  circuit  be  cut  out.     (4  min.) 

11.  The  secondary  of  a  transformer  and  the  load  it  supplies 
with  current  are  in  series.     The  e.m.f.  impressed  on  the  secondary 
is  100  and  its  resistance  and  reactance  are  0.02  and  0.085  ohms. 
The  current  is  150  amperes,  lagging  45  degrees  behind  the  impressed 
e.m.f.     Required  the  voltage  on  the  load  and  its  power  factor. 
(5  min.) 

12.  For  three  circuits  in  series  the  resistances  are  100,  75  and 
30  and  the  inductances  1,  0.3  and  0;  the  frequency  is  60;  required 
the  lag  for  each  circuit  and  for  the  system;  also  the  power  factor 
for  the  system.     (10  min.) 

13.  A  50-volt  incandescent  lamp  taking  J  ampere  is  placed  on 
a  100-volt  circuit;   what  must  be  the  value  of  L  in  a  choke  coil 
placed  in  series  with  this  lamp  to  give  the  right  voltage,  if  the 
resistance  of  the  coil  be  negligible  and  if  the  frequency  be  100? 
What  will  be  the  lag,  the  power  factor  and  the  efficiency  of  the 
combination?     (5  min.) 

14.  The  resistances  of  three  circuits  in  series  are  12,  8  and  5 
ohms;   the  inductances  of  the  first  two  are  0.1  and  0.15,  and  the 
tangent  of  the  angle  of  lag  of  the  third  is  J,  w  is  200.     With  20 
amperes  flowing  through  the  circuit,  find  the  e.m.f.  between  the 
terminals.     (5  min.) 

15.  A   choke  coil  to  turn  down  a  55-watt,   110-volt  lamp, 
reducing  the  pressure  to  50  and  25  volts,  is  to  be  designed.     What 
must  be  the  inductive  e.m.f. 's,  neglecting  the  resistances  of  the 
coil?     Required  also  the  power  factors.     (7  min.) 

16.  A  circuit  having  r,  5,  and  x,  2,  is  fed  by  a  transmission  line 
of  1  ohm  resistance.     With  a  generator  e.m.f.  of  1000  volts,  what 


INDUCTIVE  CIRCUITS  IN  SERIES  AND  PARALLEL     61 

will  be  the  e.m.f.  on  this  circuit?     What  per  cent  will  the  drop  in 
the  line,  from  no  load  to  full  load  be,  of  this  e.m.f.?     (4  min.) 

17.  A  machine  takes  30  volts  and  12  amperes;    required  the 
inductive  e.m.f.  of  a  choke  coil  to  be  placed  in  series  with  it  on  a 
100-volt  circuit,  if  the  resistance  of  the  choke  coil  and  the  induc- 
tance of  the  machine  be  negligible.     (3  min.} 

18.  A  soldering  iron  built  for  50  volts  and  3  amperes  is  to  be 
used  on  a  100-volt,  60-period  circuit,  with  a  choke  coil.     If  the 
resistance  of  the  choke  coil  is  1  ohm,  required  the  e.m.f.  around 
the  choke  coil,  the  inductive  e.m.f.  and  the  inductance.     (4  min.) 

19.  Required  the  inductive  e.m.f.  to  be  given  by  a  0.2-ohm 
choke  coil  so  that  the  e.m.f.  of  a  circuit  may  be  cut  from  100  to 
40  volts,  for  a  10-ampere  current.     What  is  the  power  factor  and 
the  efficiency?     (4  min.) 

20.  Required  the  e.m.f.  around  a  3-ohm  choke  coil  to  be  used 
in  series  with  a  50-volt,  25-candle-power  lamp,  taking  1  watt  per 
candle-power  for  use  on  a  110-volt  circuit.     Required  also  the 
efficiency.     (4  min.) 

21.  A  motor  takes  50  amperes,  lagging  30  degrees.     It  is  fed 
through  a  circuit  of  1  ohm  resistance;  what  must  be  the  generator 
e.m.f.  to  give  1000  volts  at  the  motor?     What  is  the  drop  at  the 
motor  terminals  from  no  load  to  full  load?     (3  min.) 

22.  Two  circuits  are  connected  in  series  upon  a  200-volt  e.m.f. 
The  resistance  and  reactance  of  the  first  are  0.50  and  0.75  ohms, 
and  a  current  of  20  amperes  lagging  30  degrees  behind  the  e.m.f. 
of  the  second  circuit  is  flowing;    required  the  e.m.f. 's  on  both 
circuits.     (This  problem  is  used  on  transformer  work.     The  first 
circuit  is  the  transformer  secondary,  the  second  the  load.)     (4  min.) 

23.  The  secondary  coil  of  a  transformer  and  the  circuit  which  it 
supplies  are  in  series.     The  e.m.f.  impressed  upon  the  secondary 
is  200  volts,  and  its  resistance  and  reactance  are  0.2  and  0.5  ohms. 
The  current  is  50  amperes,  and  lags  30  degrees  behind  the  terminal 
e.m.f.  of  the  transformer.     Required  the  voltage  on  the  load  and 
the  impedance  drop  in  the  secondary.     (3  min.) 

24.  An  induction  motor  taking  5000  kilowatts  at  10,000  volts 
and  85%  power  factor  is  supplied  through  a  line  having  a  resist- 
ance of  1.2  ohms,  and  a  reactance  of  0.3  ohm.     What  must  be  the 
generator  e.m.f. ;  also  the  drop  in  volts  at  the  motor  from  no  load 
to  full  load;  and  the  regulation,  that  is,  the  per  cent  which  the 
above  drop  is  of  the  full-load  voltage.     (6  min.) 


62  ELECTRICAL  ENGINEERING  PROBLEMS 

25.  A  7-ampere  series-arc  lighting  system  is  operated  upon  a 
6600-volt  circuit.     The  volts  per  lamp  are  75  and  the  power  factor 
of  the  lamps  is  83%.     The  current  is  maintained  constant  by  a 
75-light  inductive  regulator  having  3.0  ohms  resistance.     What 
will  be  the  voltage  across  its  terminals  when  the  full  load  of  lamps 
is  on;  also  for  40  lights,  and  for  5  lights?     (10  min.) 

26.  Construct  the  diagram  for  a  rotary  converter,  reactance 
coil  and  line  in  series.     The  converter  is  taking  120  amperes, 
lagging  30  degrees,  and  its  terminal  e.m.f.  is  1000  volts.     The 
resistances  of  the  coil  and  line  are  0.4  and  0.8  ohms,  and  their 
reactances  1.5  and  0.2  ohms.     Find  the  generator  e.m.f.,  the  power 
factor  of  the  system  and  the  voltage  at  the  end  of  the  line.     (6 
min.) 

27.  Find  graphically  the  sum  of  the  following  currents  and 
the  tangent  of  the  lag  of  the  resultant: 

30  amperes,  lagging  15  degrees, 
20  amperes,  lagging  45  degrees, 
10  amperes,  leading  30  degrees, 
5  amperes,  in  phase.  (4  min.) 

28.  An  induction  motor  taking  50  amperes  and  with  a  power 
factor  of  0.7  has  in  parallel  with  it  200  ^-ampere  lamps  in  parallel. 
Find  the  total  current.     (3  min.) 

29.  At  a  certain  point  in  a  circuit  there  are  50  amperes  flowing 
with  a  power  factor  of  86.6%.     20  amperes  is  then  taken  off  for 
a  group  of  incandescent  lamps  while  the  balance  passes  on  to  an 
induction  motor.     What  current  does  the  motor  take  and  at  what 
power  factor?     (3  min.) 

30.  Two  parallel  circuits  on  a  2200-volt  system  have  resistances 
of  5  and  10  and  reactances  of  5  and  20;  required  the  current  and 
lag.     (7  min.) 

31.  Fifteen  50-watt,  100-volt  lamps  are  in  parallel  on  a  100- 
volt  system  and  in  parallel  with  a  circuit  of  5  ohms  and  0.005 
henry;  the  frequency  is  120.     What  is  the  current,  the  angle  of 
lag  and  the  power  factor?     (8  min.) 

32.  Ten  enclosed  arc  lamps  in  parallel,  each  taking  6  amperes 
and  70  volts  active  e.m.f.,  and  with  choke  coils  in  series,  are 
in  parallel  with  250,   110-volt,  0.55-ampere  incandescent  lamps. 
What  will  be  the  total  current?     (6  min.) 


INDUCTIVE  CIRCUITS  IN  SERIES  AND  PARALLEL     63 

33.  On  a  10-ampere  circuit  is  to  be  placed  a  group  of  five  1- 
ampere,  50-volt  lamps  in  parallel.     To  by-pass  the  remainder  of 
the  current  a  choke  coil  (resistance  negligible)  is  to  be  used  in 
parallel  with  the  lamps.     For  what  current  must  this  coil  be 
designed?     (3  min.) 

34.  Two  circuits  in  parallel  have  respectively  TI  =  12;  tan  <fo  = 
1.25;  r2  =  30;  and  L2  =  0.015.     What  will  be  the  angle  of  lag  of 
the  resultant  current  and  what  will  be  its  value  if  E  be  117  volts 
and  co  be  500?     (8  min.) 

35.  Two  parallel  circuits  on  a  5000-volt  system  have  resistances 
of  20  and  50;    also  the  power  factor  of  the  first  is  0.447,  and  a 
current  of  40  amperes  flows  in  the  second  circuit;    required  the 
reactance  of  this  circuit  and  the  total  combined  current.     (8  min.) 

36.  Two  motors  are  running  in  parallel  on  a  500-volt  system. 
One  has  an  equivalent  resistance  of  5  and  a  power  factor  of  89.4%. 
The  combined  current  is  110  amperes  and  lags  30  degrees.     Find 
the  current  in  each  motor  and  the  power  factor  of  the  second 
motor.     (5  min.) 

37.  If  the  two  circuits  of  problem  1  are  placed  in  parallel  on  a 
220-volt  circuit,  and  the  first  circuit  takes  25  amperes,  determine 
the  frequency  and  the  current  in  the  other  circuit.     (7  min.) 

38.  Using  the  notation  of  problem  3,  but  with  the  circuits  in 
parallel  and  carrying  currents  Ii  and  I2,  draw  the  diagram  and  state 
the  solution  for  six  cases  as  follows:    To  find  Xi,  x2,  I;   r2,  Xi,  I; 
r2,  xb  x2;  E,  Ii,  I2;  E,  I,  I2;  E,  xx,  x2.     (20  min.) 

39.  An  induction  motor  taking  10  amperes  with  a  power  factor 
of  86.6%  is  placed  on  the  same  line  with  30  incandescent  lamps, 
in  parallel,  each  taking  \  ampere;   find  the  current  in  the  line. 
Required  also  the  power  factor.     (3  min.) 

40.  A  projection  lamp  taking  15  amperes  at  40  volts  and  with 
a  power  factor  of  85%  is  to  be  used  in  parallel  with  a  load  of  lamps 
on  a  110-volt  system.     Compare  the  currents  in  the  feeder  with 
a  transformer  and  with  a  choke  coil  used  to  reduce  the  e.m.f. 
(a)  With  200  J-ampere  lamps  in  parallel,     (b)  With  20  J-ampere 
lamps  in  parallel.     Which  would  you  use,  transformer  or  coil? 
(7  min.) 

41.  On  a  certain  circuit  are  an  induction  motor,  synchronous 
converter  and  a  load  of  incandescent  lamps.     The  motor  has 
equivalent  resistance  and  inductance  of  5  and  0.01,  the  converter 
3  and  0.015,  while  the  combined  resistance  of  the  lamps  is  6.25. 


64  ELECTRICAL  ENGINEERING  PROBLEMS 

If  the  frequency  is  60  and  the  motor  is  taking  16  amperes,  required 
the  currents  in  the  other  two  circuits,  the  whole  current  and  its 
power  factor  and  the  necessary  e.m.f.  (10  min.) 

42.  Two  parallel  circuits  have  resistances  of  20  and  30  ohms 
and  reactances  of  10  and  40;   find  the  e.m.f.  necessary  to  give  a 
combined  current  of  100  and  how  this  current  divides  between  the 
circuits.     (8  min.) 

43.  The  load  on  a  transformer  is  made  up  of  500,  200-ohm 
incandescent  lamps  and  an  induction  motor  circuit  having  an 
equivalent  resistance  of  0.25  and  a  power  factor  of  70.7%.     The 
current  is  500.     Find  the  e.m.f.,  the  current  in  each  circuit  and 
the  power  factor  of  the  system.     (9  min.) 

44.  The  resistance  of  two  circuits  in  parallel  are  16  and  20  ohms 
and  their  inductances  0.08  and  0.12  henries;  the  frequency  is  80; 
required  the  e.m.f.  necessary  for  a  total  current  of  10  amperes. 
(9  min.) 

45.  Three  parallel  inductive  circuits  of  3  ohms  resistance  each 
have  power  factors  of  0.2,  0.4  and  1.     The  total  current  is  200 
amperes.     Required  the  e.m.f.  and  power  factor  of  the  current. 
(4  min.) 

46.  Two  induction  motors  are  run  on  the  same  circuit  and  take 
together  400  amperes  with  a  power  factor  of  0.75;  one  motor  has 
a  power  factor  of  0.70  and  an  equivalent  resistance  of  0.8;   the 
current  taken  by  the  other  is  250.     Find  the  e.m.f.  and  the  other 
current;  also  the  other  power  factor.     (5  min.) 

47.  If  the  3  circuits  of  problem  6  are  placed  in  parallel  and  the 
combined  current  is  50  amperes,  find  the  e.m.f.  and  the  power 
factor  of  the  system.     (8  min.) 

48.  Four  parallel  circuits  have  resistances  of  20,  10,  x  and  1, 
and  inductances  of  0,  0.025,  0.12  and  y;  the  e.m.f.  is  200,  perio- 
dicity, 90,  and  the  currents  in  the  third  and  fourth  circuits,  2.6  and 
1.6.     Find  the  currents  in  the  first  and  second,  the  total  current 
and  its  lag,  and  the  equivalent  resistance  and  inductance  for  the 
whole  circuit.     (10  min.) 

49.  Two  parallel  circuits  have  0.2  and  0.03  henries  and  0  and  10 
ohms  respectively;  a  circuit  in  series  with  them  has  0.05  henry  and 
8  ohms;   the  current  through  the  10-ohm  circuit  is  3  amperes  at 
60  periods;  required  the  e.m.f.  around  each  circuit  and  around  the 
whole;  also  the  two  unknown  currents  and  the  lag  for  the  system 
as  a  whole.     (10  min.) 


INDUCTIVE  CIRCUITS  IN  SERIES  AND  PARALLEL     65 

50.  Required  the  inductive  e.m.f.  of  a  choke  coil  to  be  placed 
in  series  with  a  projection  arc  lamp  taking  15  amperes  at  35  volts 
when  used  on  a  110-volt  circuit,  the  resistance  of  the  coil  being 
J  ohm.     If  60  110-volt,  J-ampere  lamps  are  in  parallel  with  the 
above,  and  with  each  other,  what  will  be  the  total  current?     (6 
min.) 

51.  A  5-ampere  arc  lamp  is  to  be  used  on  a  7.5-ampere  constant- 
current  line.     Its  power  factor  is  86.6%  and  its  terminal  e.m.f. 
72  volts.     A  choke  coil  with  a  resistance  of  2  ohms  is  to  by-pass 
the  remaining  current.     Construct  the  diagram  and  determine 
the  current  in  the  coil  and  the  power  factor  of  the  combination. 
(7  min.) 

52.  It  is  required  to  measure  the  exciting  or  no-load  current 
of  a  J-kw.  transformer  on  a  120-volt  circuit.     In  order  to  adjust 
the  voltage  a  voltmeter  with  800  ohms  resistance  and  negligible 
reactance  is  in  parallel  with  the  transformer  so  that  the  ammeter 
reads  its  current  also.     The  total  current  is  0.45  ampere  with  a 
power  factor  of  79%.     What  is  the  exciting  current  and  its  power 
factor?     (3  min.) 


CHAPTER  VI 
CAPACITY  AND  INDUCTIVE  CIRCUITS,  RESONANCE 

1.  A  200-microfarad  condenser  is  connected  through  a  resist- 
ance of  50  ohms  to  a  100-volt  circuit  having  o>  =  400  (frequency 
63.7).     Construct  the  triangle  and  determine  the  current  and 
power  factor.     (2  min.) 

2.  The  same  condenser  is  connected  through  a  resistance  of 
2  ohms  to  a  1000-volt  circuit  of  the  same  frequency.     Construct 
the  triangle  and  find  the  current  and  power  factor.     (2  min.} 

3.  Two  circuits  have  resistances  of  5  and  20  ohms,  and  their 
capacities  are  respectively  100  microfarads  and  infinity.     If  they 
are  in  series  on  a  200-volt  circuit  with  w  =  400,  find  the  current 
and  power  factor.     (2  min.) 

4.  A  circuit  with  15  ohms  resistance  and  0.02  henry  inductance 
connects  a  500-volt  alternator  to  a  120-microfarad  condenser; 
co  is  500;   find  the  current  and  the  angle  of  lag  or  lead.     (4 
min.) 

5.  A  condenser  of  12.5  microfarads  is  fed  through  a  resistance 
of  100;  the  maximum  current  is  1  and  co  is  400.     Plot  sine  curves 
of  current  and  condenser  e.m.f.,  and  hence  determine  the  curve  of 
impressed   e.m.f.     Construct  the  vector  parallelogram   for  these 
three  curves.     (10  min.) 

6.  In  problem  5  replace  the  resistance  with  an  equal  impedance 
in  which  the  power  factor  is  0.707.     Plot  the  curves  of  active, 
induced  and  condenser  e.m.f.  and  from  them  determine  the  im- 
pressed e.m.f.  curve.     (15  min.) 

7.  In  problem  5  what  reactance  would  have  to  be  inserted  to 
bring  the  current  into  phase  with  the  impressed  e.m.f.?     Also  for 
the  circuits  of  problems  4  and  6  of  Chapter  V?     (2  min.) 

8.  Given  for  a  circuit:  E  =  100  sin0;   R  -  2;   L  =  0.01;  o>  = 
600;   capacity  =  0.001.     Find  the  equation  for  the  instantaneous 
currents.     (4  min.) 

9.  An  over-excited  synchronous  motor  takes  130  amperes  with 
a  power  factor  of  50%.     What  will  be  the  e.m.f.  at  its  terminals 

66 


CAPACITY  AND  INDUCTIVE  CIRCUITS,  RESONANCE     67 

when  connected  through  a  line  of  1.1  ohms  resistance  to  a  400-volt 
generator?     (3  min.) 

Note.  —  An  over-excited  synchronous  motor  or  converter  acts 
like  a  circuit  with  resistance  and  capacity;  that  is,  it  has  a  leading 
current. 

10.  Upon  a  circuit  having  co  =  400  (frequency  =  63.7),  two 
circuits  in  series  have  respectively  19  ohms  and  0.007  henry,  and 
5  ohms  and  125  microfarads,  what  e.m.f.  will  be  necessary  to  send 
1 1  amperes  through  the  combination?     What  change  in  frequency 
would  give  unity  power  factor  or  resonance?     (5  min.) 

11.  Upon  the  same  circuit  as  above,  two  circuits  are  connected 
in  series,  one  of  which  has  7.2  ohms  and  0.22  henry,  and  the  other 
5  ohms  and  32  microfarads.     What  line  e.m.f.  will  give  1000  volts 
on  the  condenser?     What  will  be  the  current  and  the  voltage  on 
the  inductive  circuit?     (5  min.) 

12.  In  problem  11  what  capacity  would  be  needed  to  give  a 
leading  current  with  90%  power  factor?     Also  to  give  unity  power 
factor  or  resonance?     Check  the  graphical  solution  by  the  formula 
for  series  resonance.     (4  min.) 

13.  An  induction  coil  has  a  resistance  of  25  ohms,  and  when 
placed  on  a  1000-volt  circuit  with  co  =  400,  passes  2  amperes. 
What  must  be  the  capacity  of  a  condenser  in  series  with  this 
circuit  to  give  resonance?     If  the  same  line  voltage  were  main- 
tained, what  current  would  flow,  and  what  would  be  the  voltage 
around  the  induction  coil  and  also  that  at  the  terminals  of  the 
condenser?     Give  the  graphical  solution  and  check  analytically. 
(6  min.) 

14.  Construct  the  sine  curves  of  e.m.f.  around  the  condenser 
and  the  inductance  coil  in  problem  13,  and  from  them  derive  the 
curve  of  impressed  e.m.f.     Compare  the  result  with  that  obtained 
in  problem  13.     (15  min.) 

15.  In  problem  10,  if  the  circuits  are  placed  in  parallel,  find  the 
necessary  e.m.f.     (6  min.) 

16.  Three  circuits  in  parallel  have  respectively  8,  12  and  3 
ohms,  0,  0.06  and  0  henries  and  200,  infinity  and  300  microfarads 
capacity;  co  is  500.     If  200  volts  are  applied,  required  the  current, 
the  phase  angle  and  the  power  factor.     (10  min.) 

17.  Two  circuits  are  in  parallel  on  a  100-volt,  60-period  system; 
one  has  10  ohms  resistance  and  3  ohms  reactance;  the  other  has 
resistance  and  60-degree  lead.     Determine  what  its  capacity  and 


68  ELECTRICAL  ENGINEERING  PROBLEMS 

resistance  must  be  in  order  that  the  current  shall  be  brought  into 
phase  with  the  e.m.f.  so  as  to  produce  resonance.     (6  min.) 

18.  In  problem  17  if  the  resistance  of  the  condenser  circuit  be 
eliminated,  what  will  the  necessary  capacity  become?     (2  min.) 

19.  In  problem  18  if  the  resistance  in  the  first  circuit  be  reduced 
to  2,  what  will  the  necessary  capacity  become?     (4  min.) 

20.  Given  an  enclosed  arc  lamp  taking  75  volts  at  the  arc 
and   7  amperes,  and  used  on  a   100-volt  circuit;    required  the 
capacity  that  would  have  to  be  used  in  parallel  to  make  the 
power  factor  equal  to  1 ;   also  the  current  under  these  conditions. 
(4  min.) 

21.  An  induction-motor  current  of  25  amperes  has  a  power 
factor  of  80%;    the  e.m.f.  is  947  and  the  frequency  60.     If  the 
wattless  current  required  were  to  be  supplied  by  a  condenser  in 
parallel,  its  circuit  having  no  resistance,  what  would  be  its  capacity? 
(3  min.) 

Note.  —  If  the  wattless  current  is  supplied  by  a  condenser,  the 
power  factor  becomes  unity. 

22.  If ,  in  the  above,  the  frequency  were  doubled,  what  would 
be  the  capacity  needed?     Also  if  the  e.m.f.  were  halved  with  the 
frequency  at  60?     (1  min.) 

23.  A  generator  giving  4000  volts  feeds  a  5-microfarad  con- 
denser through  an  inductive  circuit  in  which  r  is  10,  L  is  1.25  and 
co  is  400.     Required  the  e.m.f.  on  the  condenser.     (2  min.) 

24.  100  volts  at  96  periods  is  applied  to  a  system  consisting  of 
two  parallel  circuits  having  respectively  0.008  and  0.02  henry,  0 
and  10  ohms  and  capacities  of  125  microfarads  and  infinity  re- 
spectively.    In  series  with  these  is  another  circuit  having  0.005 
henry  and  5  ohms;    required  the  currents,  unknown  e.m.f. 's  and 
lag  for  the  system.     (12  min.) 

25.  What  must  be  the  capacity  of  a  condenser  to  be  placed  in 
parallel  with  the  inductance  coil  of  problem  13  in  order  to  give 
resonance  on  a  circuit  with  co  =  400,  and  what  line  current  would 
have  to  be  supplied  to  obtain  40  amperes  maximum  through  the 
inductance,  coil?     What  would  be  the  maximum  impressed  e.m.f.? 
Check  the  results  as  obtained  graphically  by  the  proper  formula. 
(8  min.) 

26.  Construct  the  sine  curves   which   are  represented  by  the 
vectors  of  problem  25  and  prove  the  numerical  relation  of  their 
ordinates  for  three  points.     (20  min.) 


CAPACITY  AND  INDUCTIVE  CIRCUITS,  RESONANCE     69 

27.  Given  a  circuit  of  6.52  ohms  resistance  and  80  microfarads 
capacity,  to  which  an  e.m.f .  of  130  periods  and  400  volts  is  applied. 
Find  the  value  of  L  which  makes  the  current  a  maximum,  and  then 
the  values  of  the  current  and  lag  or  lead  for  the  following  values 
of  L:  0,  0.01,  0.03,  0.04.  Plot  these  in  terms  of  L.  (30  raw.) 


CHAPTER  VII 
SINGLE-PHASE  POWER,  WATTMETERS 

1.  Required  the  power  consumed  in  each  of  the  following  cases : 
50  amperes  at  1000  volts  and  the  power  factor  0.80; 

30  amperes  at  400  volts  with  a  30-degree  lag; 

1000  volts  at  60  periods  on  a  circuit  of  10  ohms  and  0.02  henry. 

(3  min.) 

2.  A  single-phase  motor  takes  49  kilowatts  at  350  volts  with  a 
lag  of  30  degrees;  what  will  be  the  current  and  how  much  larger 
must  the  conductors  be  than  for  a  direct  current  of  the  same 
voltage?     (2  min.) 

3.  A  110-volt  generator  is  rated  at  120  kilo  volt-amperes  or  120 
kilowatts  on  an  inductionless  load.     What  would  be  its  rating  on 
the  following  loads:   (a)  A  power  factor  of  0.9;   (b)  a  power  factor 
of  0.7;    (c)  current  lagging  30  degrees;    (d)  current  leading  60 
degrees.     (1  min.) 

4.  The  station  voltmeter  reads  1100,  the  ammeter  reads  150 
and   the   wattmeter   reads   120  kilowatts;     required  the   power 
factor  of  the  system  and  the  lag;   also  if  the  generator  is  fully 
loaded,  what  would  be  its  kilowatt  capacity  at  unity  power  factor. 
(2  min.) 

5.  A  motor  is  taking  100  amperes  (maximum)  at  250  volts  and 
with  a  lag  of  30  degrees.     Plot  sine  curves  of  e.m.f.  and  current 
and  by  multiplication  of  the  ordinates  derive  and  plot  a  curve 
of  instantaneous  power  values.     Integrate  this  curve  (measure  its 
area)  and  thus  check  the  formula  for  power  in  an  inductive  circuit. 
(20  min.) 

6.  A  motor  circuit  connected  to  a  100- volt  alternating-current 
system  is  taking  25  amperes;   the  wattmeter  reads  1.8  kilowatts; 
plot  curves  of  e.m.f.  (sine),  current  and,  derived  from  these,  watts, 
and  check  by  integrating  (measuring  area)  the  watt  curve.     Take 
1  large  division  equal  to  30  degrees.     (20  min.) 

7.  An  inductance  coil  on  a  500-volt,  maximum,  circuit  has  r,  5, 
and  takes  10  amperes,  <o  is  400;  plot  the  e.m.f.  current  and  hence 

70 


SINGLE-PHASE  POWER,  WATTMETERS  71 

the  power  curves,  and  find  from  them  the  power  consumption. 
What  are  the  power  and  wattless  components  of  the  cur- 
rent? Take  one  large  division  of  the  paper  equal  to  30  degrees. 
(20  raw.) 

8.  Plot  the  curves  and  prove  graphically,  that  is  by  measure- 
ment of  area,  that  for  the  following  case  the  watts  equal  IE  cos  $>; 
maximum  current,  126;    maximum  e.m.f.,  140;    lag,  36  degrees. 
Take  one  large  division  equal  to  30  degrees.     Where  is  the  axis 
of  the  power  curve  relative  to  the  axis  of  the  e.m.f.  curve?     (20 
min.) 

9.  A  lamp  and  an  induction  coil  are  in  series  on  a  120-volt 
system.     The  coil  cuts  down  the  voltage  on  the  lamp  to  70  at 
which  it  takes  0.15  ampere.     The  resistance  of  the  coil  is  50  ohms. 
Plot  curves  of  instantaneous  power  values  for  the  lamp,  the  coil 
and  the  combined  load.     (30  min.) 

10.  Draw  a  sine  curve  for  an  e.m.f.  of  100  volts  maximum,  and 
for  a  current  of  50  amperes  with  a  power  factor  of  70.7%.     From 
these  plot  the  power  curve.     Repeat  for  the  same  current  with  a 
power  factor  of  0.5,  and  with  zero  power  factor.     (40  min.) 

11.  The  cost  of  a  certain  10,000-kilovolt-ampere  power-plant 
complete  is  $80.00  per  kilo  volt-ampere.     If  the  current  supplied 
by  this  plant  lags  30  degrees  what  will  be  the  available  output  in 
kilowatts,  and  what  is  the  decrease  in  the  value  of  the  plant  as 
compared  with  the  same  plant  at  unity  power  factor?     (1  min.) 

12.  10,000  kilowatts  are  transmitted  100  miles  at  80,000  volts 
over  a  line  which  has  a  resistance  of  64  ohms.     The  power  factor 
is  70%.     If  the  power  costs  the  power  company  0.6  cents  per 
kilowatt-hour  and  is  used  10  hours  a  day  for  300  days  in  the  year, 
what  would  be  the  annual  saving  on  copper  loss  if  the  power  factor 
could  be  increased  to  unity?     (3  min.) 

13.  Separating  the  two  coils  of  a  dynamometer  ammeter  and 
sending  two  currents  differing  in  phase  through  them,  the  instru- 
ment reads  50.3.     Ammeters  in  these  two  circuits  read  7.35  and 
11.47  amperes;    what  is  the  phase  angle  between  the  currents? 
(2  min.) 

14.  Required  the  maximum  inductance  that  the  shunt  coil  of 
a  wattmeter  can  have  if  its  resistance  is  1500,  in  order  that  the 
error  with  a  100-period  current  shall  not  exceed  £  of  1%,  the 
maximum  lag  in  the  circuit  to  be  measured  being  45  degrees. 
(2  min.) 


72  ELECTRICAL  ENGINEERING  PROBLEMS 

15.  A  wattmeter  is  to  measure  power  in  a  circuit  having  a 
maximum  angle  of  lag  of  36  degrees;  the  periodicity  is  60,  and  the 
resistance  of  the  wattmeter  shunt  is  1000  ohms.  How  small  must 
the  value  of  L  be  in  the  wattmeter  in  order  to  get  the  results 
correct  to  within  \  of  1%?  The  same  problem,  but  with  an  angle 
of  lag  of  60  degrees.  (6  min.) 


CHAPTER  VIII 
POLYPHASE  SYSTEMS  AND  POWER 

1.  A  2-phase,  3-wire  system  carries  30  amperes  in  each  outside 
wire;   what  will  be  the  current  in  the  common  return?     (|  min.) 

2.  A  2-phase,  3-wire  system  is  supplying  100  kilowatts  at  500 
volts  and  with  80%  power  factor;  what  will  be  the  current  in  the 
third  wire?    What  is  the  e.m.f.  between  the  outside  wires?     (2 
min.) 

3.  A  3-phase,   Y-connected  system  has  4000  volts  between 
wires.     The  neutral  point  is  grounded.     For  what  voltage  must 
the  insulation  of  the  windings  be  designed?     (J  min.) 

4.  A  3-phase,  Y-connected  system  with  a  fourth  or  neutral 
wire  is  designed  to  use  110-volt  lamps  between  the  neutral  and  the 
outsides;  what  will  be  the  e.m.f.  between  the  three  lines?     (J  min.) 

5.  A  3-phase,  delta-connected  system  carries  currents  of  50 
amperes  in  the  delta  connections;  what  is  the  current  in  the  lines? 
(i  min.) 

6.  With  300  amperes  in  the  line  of  a  delta-connected,  3-phase 
system,  what  current  will  flow  in  the  mesh?     (J  min.) 

7.  A  2-phase,  3-wire  system  is  out  of  balance;  the  currents  are 
100  and  90  amperes,  with  an  angle  of  85  degrees  between  them. 
What  is  the  current  in  the  third  wire?     (2  min.) 

8.  The   currents   in   an   unbalanced   delta-connected   3-phase 
system  are  80,  90  and  100  amperes;  what  are  the  line  currents  and 
what  are  their  phase  relations  (instead  of  120  degrees)?    Solve 
graphically.     (5  min.) 

9.  The  currents  in  the  lines  of  a  mesh-connected,  4-phase 
system  are  400  amperes;  what  will  be  the  currents  in  the  meshes? 
With  1000  volts  between  adjacent  lines,  what  will  be  the  e.m.f. 
between  opposite  ones?     (1  min.) 

^10.  In  the  balanced  2-phase,  3-wire  system  of  problem  2,  the 
Series  coil  of  a  wattmeter  is  put  in  the  third  wire  while  the  pressure 
coil  is  across  the  outside.  What  will  the  instrument  read  and 
why?  (2  min.) 

11.   By  the  revolution  of  a  vector  diagram  construct  the  three- 

73 


74      ELECTRICAL  ENGINEERING  PROBLEMS 

current  curves  of  a  100-ampere,  3-phase,  Y-connected  generator; 
also  draw  circuit  diagrams  showing  the  values  and  directions  of 
the  instantaneous  currents  in  each  circuit  for  a  =  0°,  30°,  45°, 
60°  and  90°  and  observe  that  the  sum  of  the  currents  is  zero  in 
_each  case.  (Hand  in  the  vector  diagram.)  (10  min.) 

12.  Let  the  armature  of  the  last  problem  be  delta-connected 
with  100  amperes  in  the  outside  circuits;    construct  the  vector 
diagram  for  the  armature  and  line  currents  and  by  its  revolution 
the  sine  curves  for  all  the  currentsv    Use  the  conventions  that 
currents  are  positive  in  the  mesh  when  counterclockwise  and  in 
the  line  when  directed  away  from  the  junctions.     Draw  the  five- 
circuit  diagrams  as  in  problem  11.     (Hand  in  the  vector  diagrams.) 
(20  min.) 

13.  A   100-volt,   delta-connected,   three-phase  system  has  in 
each  branch  resistance  and  reactance  as  follows:  circuit  (a),  3  and 
3  ohms;    circuit  (b),  6  and  0  ohms;    circuit  (c),  1  and  5  ohms. 
Construct  the  diagram  and  determine  the  currents  in  the  meshes 
and  leads.     (10  min.) 

14.  A  200-volt,  delta-connected  system  has  currents  of  27,  43 
and  18  with  power  factors  of  100,  86.6  and  50%.     Determine  the 
currents  in  the  leads  and  their  phase  relations  (tangents)  to  the 
e.m.f.'s.     (10  min.) 

15.  Each  circuit  of  a  120-volt,  Y-connected,  3-phase  system 
has  a  resistance  of  3  and  a  reactance  of  6  ohms.     Construct  the 
diagram  and  determine  the  line  currents  and  their  power  factors. 
(8  min.) 

16.  The  three  circuits  of  a  150-volt,  Y-connected  system  are 
made  up  of  groups  of  incandescent  lamps  having  resistances 
respectively  of  5,  8  and  10  ohms.     Determine  the  current  in  each 
lead  and  the  voltage  to  neutral  around  each  circuit.     (15  min.) 

17.  The  three  circuits  of  problem  13  are  connected  Y  instead 
of  delta;  required  the  current  and  voltage  of  each  and  the  phase 
angles  (tangents)  between  the  currents  and  line  e.m.f.'s.     (20  min.) 

18.  If  in  problem  16  a  fourth,  or  neutral,  wire  is  used,  what 
will  be  the  current  in  each  of  the  four  lines?     (4  min.) 

19.  If  in  problem  17  a  fourth  wire  is  used,  what  will  be  the 
current  in  each  of  the  four  lines?     (8  min.) 

20.  A  two-phase,  three-wire  system  takes  200  amperes,  with  a 
power  factor  of  86.6  in  each  phase,  from  a  100-volt  generator  and 
is  supplied  through  lines,  each  of  which  has  a  resistance  of  0.1  ohm. 


POLYPHASE  SYSTEMS  AND  POWER  75 

What  will  be  the  voltage  on  each  circuit  and  the  phase  angle 
(tangent)  between  it  and  the  current?  Note  the  effect  of  the 
common  return  in  unbalancing  the  circuit.  (8  min.) 

21.  A  4- wire,   3-phase   system  is  supplying   150  amperes  at 
unity  power  factor  to  synchronous  motors  connected  delta,  and 
85  amperes  each  to  three  groups  of  incandescent  lamps  connected 
to  the  fourth  wire.     Find  the  current  in  each  wire  and  the  motor 
voltage  if  the  lamps  are  for  125  volts.     (8  min.) 

22.  A  3-phase,  delta-connected  system  has  5  ohms  non-induc- 
tive resistance  in  each  mesh  circuit.     There  are  also  3  ohms  in  each 
main  circuit  outside  of  the  delta.     With  110  volts  applied  to  the 
system,  what  will  be  the  current  in  each  circuit  and  the  e.m.f. 
around  the  delta?     (6  min.) 

23.  Same  as  problem  22,  except  that  2  ohms  reactance  is  added 
to  each  of  the  mesh  circuits.     (8  min.) 

24.  Same  as  problem  23,  except  that  2  ohms  reactance  is  added 
to  each  of  the  main  circuits.     (8  min.) 

25.  Given  a  ring-wound  armature  placed  in  a  2-pole  field  and 
tapped  at  0,  90,  120,  180,  240  and  270  degrees.     With  250  volts 
across    opposite   brushes,   required  the  voltage   across  adjacent 
brushes  (90  degrees  apart)  of  2-phase  and  across  the  3-phase  lines. 
If  the  generator  is  to  give  100  kilowatts,  what  will  be  the  current 
in  the  armature  conductors  for  the  single,  2-phase  and  3-phase? 
Also  what  in  the  line?     (8  min.) 

26.  If  the  armature  of  problem  25  is  wound  with  2  flat  wires 
50  X  400  mils  and  600  circular  mils  per  ampere  be  allowed,  what 
would  be  the  capacity  as  single,  2  and  3-phase  generators?     (6 
min.) 

27.  Find  the  wire  area  and  the  B.  &  S.  size  for  the  lines  to 
transmit  430  kilowatts  at  22,000  volts,  single-phase,  3-phase  and 
2-phase,  4-wire  and  3-wire.     Using  the  B.  &  S.  numbers  required, 
at  20  cents  per  pound,  find  the  cost  per  mile  of  circuit  in  each  case. 
Take  1200  circular  mils  per  ampere  as  giving  a  satisfactory  drop. 
(15  min.) 

28.  Given  a  3-phase  motor  with  400  volts  between  the  lines, 
power  factor  of  85%,  efficiency  of  80%  and  with  10  horse-power 
delivered;  required  the  current  in  the  lines  and  also  in  the  con- 
ductors of  the  delta-wound  armature.     (3  min.) 

29.  Two  wattmeters  are  placed  in  the  outside  wires  of  a  bal- 
anced 2-phase,  3-wire  system  feeding  two  similar  non-inductive 


76  ELECTRICAL  ENGINEERING  PROBLEMS] 

circuits  with  100  volts;  when  normally  connected,  each  of  the 
instruments  indicates  500  watts,  but  when  the  shunt  coils  are 
interchanged,  each  reads  50  watts.  Draw  a  diagram  to  represent 
this  case  and  find  the  currents.  (5  ram.) 

30.  The  two  wattmeters  on  a  2-phase  system  indicate  1500 
and  200  watts;  the  e.m.f.'s  are  both  100  volts  and  are  90  degrees 
apart;   the  currents  are  18  and  3  amperes.     Draw  a  diagram  for 
the  case  and  determine  the  lags  (tangents)  of  the  currents.     (7 
ram.) 

31.  A  2-phase,  3-wire  system  with  e.m.f.'s  of  400  volts,  90 
degrees  apart  has  currents  of  25  amperes  in  each  phase  lagging 
30  degrees;  what  total  power  would  the  wattmeter  readings  show 
with  the  two  wattmeters  placed  normally,  and  also  with  their 
series  coils  both  in  the  common  wire;  solve  graphically.     (8  ram.) 

32.  In  problem  31,  if  one  current  were  30  amperes,  what  would 
the  sums  of  the  wattmeter  readings  become?     (10  ram.) 

33.  Three  wattmeters   connected  in  the   delta   circuits  of^  a 
3-phase  system  show  1120,  1540  and  1780  watts,  respectively,  the 
voltages  are  each  440  and  are  120  degrees  apart,  the  currents  are  3.2, 
4.5  and  6.3  amperes.     Construct  the  diagram  for  this  case  includ- 
ing the  line  currents  and  give  the  total  power  and  the  power  factor 
of  each  circuit.     (12  ram.) 

34.  Three  wattmeters  are  used  with  the  voltage  coils  across 
the  circuits  to  measure  the  power  in  a  220-volt,  3-phase,  Y-con- 
nected  system.     Each  reads  1250  watts  and  power  factor  meters 
show  75%  power  factor.     Construct  the  diagram  for  the  case  and 
find  the  line  currents  and  the  total  power.     (7  ram.) 

35.  The  currents  in  the  three  leads  of  a  500- volt  balanced  delta- 
connected,  3-phase  system  are  each  173.2  amperes;  the  readings 
of  the  two  wattmeters  properly  connected  to  measure  the  power  of 
the  system  are  each  75  kilowatts.     What  is  the  power  factor  of 
the  system?     Prove  by  means  of  a  vector  diagram.     (6  ram.) 

36.  With  the  same  system  as  in  problem  35,  except  that  the 
wattmeters  read  75  and  0  kilowatts,  draw  the  diagram  and  deter- 
mine the  power  factor.     (5  ram.) 

37.  Repeat  problem  35,  except  that  the  wattmeter  readings 
are  plus  60  and  minus  24  kilowatts.     (7  ram.) 

38.  What  would  be  the  readings  of  two  wattmeters  with  their 
series  coils  in  the  same  branches  as  circuits  (a)  and  (b)  of  problem 
17?     (10  ram.) 


POLYPHASE  SYSTEMS  AND  POWER  77 

39.  Repeat  problem  38,  except  that  the  wattmeters  are  in  the 
same  branches  as  circuits  (b)  and  (c).     (10  min.) 

40.  Determine  graphically  what  the  readings  on  two  watt- 
meters placed  with  their  series  coils  in  the  leads  between  the  first 
and  second  and  between  the  second  and  third  circuits  of  problem 
33  would  be.     (10  min.) 

41.  A  3-phase,  delta-connected  alternator  is  giving  750  amperes 
for  one  phase  only.     What  will  be  the  current  in  each  circuit  of 
the  alternator?     Also  what  if  a  200-ampere,  single-phase  circuit 
is  connected  to  each  of  two  phases?     (5  min.) 

42.  Show  what,  on  a  basis  of  heating,  will  be  the  capacity  of 
a  100-kilovolt-ampere,  3-phase  delta-connected  alternator  when 
run  single  phase.     Also  if  the  alternator  is  Y-connected.     (2  min.) 

43.  Repeat  problem  42,  except  that  the  load  is  to  be  two  equal 
single-phase  circuits,  one  on  each  of  two  phases.     (3  min.) 


CHAPTER  IX 
TRANSFORMERS,  GENERAL 

1.  A  10  to  1  transformer  has  the  same  length  of  turn  and  current 
density  in  both  coils.     The  secondary  resistance  is  0.01  and  the 
secondary  turns  are  80;    required  the  primary  resistance.     With 
110  volts  and  2.2  kilowatts,  given  by  the  secondary,  required  the 
primary  e.m.f.  and  current  (neglect  drop  and  magnetizing  current). 
{^  raw.) 

2.  The  resistance  of  the  two  coils  of  a  transformer  marked  for 
110-volts  secondary  are  0.3  and  0.012;  if  the  current  density  and 
length  of  turn  are  the  same,  for  what  primary  voltage  is  the  trans- 
former designed?     (1  ram.) 

3.  A  500-kv-a.,  40,000-volt  transformer  with  2000-volts  second- 
ary, has  420  secondary  turns;  what  would  be  the  area  of  this 
secondary  conductor  in  square  inches  allowing  2000  circular  mils 
per  ampere?     Also  the  resistance  if  the  mean  length  of  a  turn  is 
9.5  feet  and  the  resistance  of  a  circular-mil-foot  be  taken  as  12.5? 
Also  required  the  primary  turns,  current  and  resistances  with  same 
current  density  and  length  of  turn  in  the  two  coils.     (5  raw.) 

4.  In  problem  2,  if  2%  drop,  no  load  to  full  load  is  permitted, 
what  kilowatt  output  would  be  obtained  (neglect  leakage)  ?     What 
primary  e.m.f.  necessary  for  110  volts  on  the  secondary,  at  full 
load?     (3  raw.) 

5.  A  3-kv-a.,   10   to  1  transformer   giving  110  volts   on  the 
secondary  and  having  equal  primary  and  secondary  losses  has 
a  regulation  of  2%;  what  must  be  the  resistance  of  each  coil  and 
what  is  the  drop  in  each?    Neglect  magnetic  leakage  and  hence 
the  reactance  of  the  coils.     (3  raw.) 

6.  A  transformer  has  two  primary  coils  each  of  900  turns  and 
two  secondaries,  each  of  90  turns;   with  the  former  in  series  the 
transformer  is  suitable  for  use  on  a  2200-volt  system;   for  what 
other  e.m.f.  can  the  primary  coils  be  connected,  and  what  secondary 
e.m.f.  's  can  be  obtained  in  each  case?     (1  raw.) 

7.  A  transformer  has  2  secondary  coils  made  up  each  of  48 
turns  of  No.  8  wire,  the  length  of  the  turn  being  2  ft.  and  the 

78 


TRANSFORMERS,  GENERAL  79 

resistivity  12;  with  the  coils  in  parallel  the  transformer  is  to  give 
110  volts  at  full  load.  Using  1200  circular  mils  per  ampere, 
required  the  current  that  the  transformer  will  give  at  this  pressure; 
also  the  current  and  e.m.f.  with  the  coils  in  series;  the  primary 
winding  is  made  up  of  2  coils  each  having  240  turns  of  the  same 
mean  length  and  using  the  same  current  density;  required  the 
currents  and  e.m.f. 's  in  the  primary  with  the  coils  both  in  series 
and  in  parallel;  required  also  the  secondary  and  primary  resist- 
ances, the  primary  and  secondary  drops  and  the  regulation  (neglect 
magnetizing  current).  (10  min.} 

8.  A  4-kilowatt,  10  to  1  transformer  has  100  square  centimeters 
cross-section  of  iron  and  with  600  turns  in  the  primary  has  a  density 
at  1000  volts  of  6500;  what  would  the  density  become  if  used  on 
a  1200-volt  system?     If  the  frequency  were  decreased  10%,  what 
would  it  be?     Also  if  the  area  were  made  125.     Also  if  the  turns 
were  changed  to  500.     (5  min.) 

9.  Two  transformers  have  the  same  iron  volume  but  different 
windings,  so  that  B  is  equal  to  4000  and  6000  respectively.     Re- 
quired the  ratio,  (a)  of  the  hysteresis  losses  and  (b)  the  eddy 
current  losses.     (4  min.) 

10.  The  cross-section  and  hence  the  volume  of  the  iron  in  a 
transformer  is  increased  20%;    with  the  same  winding,   what 
changes  will  result  in  the  hysteresis  loss  and  in  the  eddy  cur- 
rent loss.     What  changes  would  then  have  to  be  made  in  the 
winding  to  bring  each  of  these  losses  back  to  its  former  value? 
(10  min.) 

11.  Without  changing  the  iron  and  e.m.f.  of  a  transformer  the 
number  of  turns  is  increased  10%.     What  changes  in  the  hysteresis 
and  eddy  current  losses  will  result?     If  in  this  transformer  the 
copper  and  iron  losses  are  equal,  what  will  be  the  percentage  change 
in  the  total  loss  (neglect  the  eddy  loss).     (8  min.) 

12.  A  50-kilo volt-ampere  transformer  has  210  watts  hysteresis 
and  32  watts  eddy  current  loss.     If  the  magnetic  density  in  this 
transformer  is  9000,  what  would  be  the  effect  on  the  hysteresis, 
eddy  and  total  iron  loss  of  increasing  it  to  11,000?     Of  reducing 
it  to  7000?     Use  1.7  for  the  hysteresis  exponent.     (7  min.) 

13.  In  problem  12  if  there  were  no  change  in  density,  but  a 
change  from  60  to  25  in  the  frequency,  what  would  the  losses 
become?     (4  min.) 

14.  If  in  problem  13  the  impressed  e.m.f.  were  kept  constant, 


80  ELECTRICAL  ENGINEERING  PROBLEMS 

what  changed  loss  would  result  from  the  combined  changes  in 
frequency  and  density?     (6  min.) 

15.  If  4750  be  the  highest  density  (maximum)  allowable  in  a 
5-kilovolt-ampere,  60-period,  10  to  1  transformer  on  1150  volts 
primary,  how  many  secondary  turns  must  be  used  if  the  area  of 
the  iron  is  112  square  centimeters?     (4  min.) 

16.  A  20-kilo volt-ampere  transformer,  designed  for  a  60-period 
circuit,  and  having  losses  as  follows :  Hysteresis,  110;  eddy  current, 
19,  and  copper  loss,  272,  is  to  be  used  on  a  140-period  circuit. 
If,  at  the  density  employed,  the  hysteresis  loss  per  cycle  varies  as 
B1-7,  what  will  be  the  efficiency  in  each  case  (neglect  the  change 
in  copper  loss).     (5  min.} 

17.  A  1500-watt  transformer  used  on  1000  volts  has  losses  as 
follows:    28  watts  hysteresis,  4.5  watts  eddy  current,  and  40.6 
watts  copper.     It  is  also  run  on  a  600-volt  circuit;  with  the  same 
current  output  and  neglecting  the  change  in  the  primary  current, 
required  the  efficiency  in  each  case.     (4  min.) 

18.  A  30  kv-a.,  1100  to  110-volt  transformer  has  a  no  load  loss 
of  165  watts,  and  primary  resistance  of  0.25;   find  the  resistance 
of  the  secondary  and  determine  the  efficiency  of  the  transformer 
at  full  load,  f ,  J  and  J  load.     Plot  these  results.     Neglect  the  effect 
of  the  exciting  current.     (10  min.) 

19.  A  2400  to  120  volt,  20-kilo  volt-ampere  transformer  has  a 
primary  resistance  of  1.9  ohms,  and  an  iron  loss  of  129  watts; 
required  the  all-day  efficiency,  if  the  load  is  as  follows: 

Number  of  hours        16      J       f      2      3      2 

Per  cent  of  full  load      0     100    75     60     50     25  (10  min.) 

20.  A  2500-watt  transformer  has  an  iron  loss  of  32  and  a  copper 
loss  of  53  watts  at  full  load;  the  primary  and  secondary  losses  are 
equal;  required  the  efficiencies  at  j,  f,  J  and  J  load;  also  the  all- 
day  efficiency  if  the  transformer  runs  2  hours  at  full  load,  4  hours 
on  %  load  and  18  hours  on  no  load.     (10  min.) 

21.  The  total  volume  of  iron  in  a  certain  transformer  is  12,400 
cubic  centimeters.     The  iron  used  is  a  silicon  steel,  having  a 
constant  of  7  X  10~u  for  watts  hysteresis  loss  per  cycle  per  cubic 
centimeter  and  an  exponent  of  1.7.     The  eddy  current  loss  coeffi- 
cient may  be  taken  as  4  X  10~15  and  the  exponent  as  2.     What  will 
be  the  hysteresis,  eddy  and  combined  iron  loss  for  B  =  10,000  per 
square  centimeter  and  60  cycles  per  second?     (5  min.) 


TRANSFORMERS,  GENERAL  81 

22.  A  10-kilo volt-ampere,  single-circuit  or  auto-transformer  is 
designed  for  220  to  110  volts  and  a  current  density  of  1000  circular 
mils  per  ampere;   what  should  the  areas  of  the  two  parts  of  the 
winding  be?     Also  for  the  following  ratios  of  transformation: 
220  to  160  volts  and  220  to  50  volts.     What  are  the  relative  weights 
of  copper  for  these  different  ratios?     (8  min.) 

23.  Determine  the  wire  areas  and  relative  weights  of  two- 
circuit  transformers  for  the  same  service  as  in  problem  22.     Also 
compare  the  weights  of  single-  and  two-circuit  transformers  for 
each  case.     (12  min.} 

24.  A     2-kilo volt-ampere     auto-transformer    is     wound    with 
No.  12  and  No.  9  wire;  for  what  primary  and  secondary  voltages 
is  it  available  if  1000  circular  mils  per  ampere  be  allowed.     Use 
the  No.  9  wire  as  secondary.     (5  min.) 

25.  A  5-kilo volt-ampere  transformer  for  changing  from  220  to 
150  volts  is  to  be  built;  with  1500  circular  mils  per  ampere,  3  feet 
length  of  turns  and  100  turns  in  the  primary,  how  many  pounds 
of  copper  would  be  saved  by  using  a  single-circuit  design?     Take 
the  weight  of  a  mil  foot  as  3.03  X  10~6  pounds  and  add  2%  (ap- 
proximate for  large  wires)  for  double-cotton-covered  insulation. 
(8  min.) 

26.  Repeat  problem  25,   except  with  a  240-volt  secondary. 
(8  min.) 

27.  In  the  Northwest  Station  of  the  Chicago  Commonwealth 
Edison  Company,  auto-transformers  are  used  to  step  up  the  power 
generated  from  4500  to  9000  volts,   the  transmission  voltage. 
Using  650  amperes  per  square  inch,  what  would  be  the  cross-section 
area  of  the  primary  and  secondary  parts  of  the  winding  of  a  6600 
kilovolt-ampere  transformer  for  this  purpose?     (2  min.) 

28.  Three  standard  2200-  to  220-volt  transformers  are  con- 
nected with  their  primaries  in  Y  on  a  2200-volt,  3-phase  system, 
while  their  secondaries  are  connected  delta- wise;  what  will  be  the 
e.m.f.  on  the  secondaries?     (1  min.) 

29.  It  is  desired  to  put  taps  on  three  10  to  1  transformers,  each 
having  80  turns  in  the  secondaries  so  that  when  the  primaries  are 
connected  in  delta  on  a   1100-volt  system,  the  secondaries  can 
be  connected  in  Y  and  give  110  volts  between  their  terminals; 
how  many  turns  from  the  neutral  must  the   taps   be   placed? 
(8  min.) 

30.  Two  standard  2200-  to  220-volt  transformers,  each  having 


82  ELECTRICAL  ENGINEERING  PROBLEMS 

150  secondary  turns,  are  to  be  used,  by  means  of  taps,  to  trans- 
form from  2200-volt,  2-phase  to  220-volt,  3-phase.  Where  must 
the  taps  be  put?  What  would  the  voltages  on  the  3-phase  system 
be  if  the  whole  secondary  were  used?  (3  ram.) 

31.  Three  transformers  to  transform  from  1000  to  100  volt, 
3-phase  are  to  have  the  primaries  connected  Y  and  the  secondaries 
connected  delta.     If  the  secondaries  have  60  turns  each,  how  many 
turns  must  the  primaries  have?     (2  min.) 

32.  Three  transformers  connected  in  delta  are  supplying  100 
amperes  to  a  three-phase  system.      What  is  the  current  in  the 
transformers?     One  of  these  transformers  burns  out  and  is  re- 
moved;  the  system  continuing  to  draw  100  amperes,  what  is  the 
current  in  the  two  remaining  transformers?     Solve  graphically. 
(10  min.) 

33.  Explain  how  two  identical  transformers  without  taps  can 
be  connected  to  give  a  3-phase  system  with  e.m.fVs  of  220,  311  and 
220  volts  when  fed  from  a  1100-volt,  2-phase  system.     What  will 
be  the  secondary  e.m.f.  and  transformation  ratios  of  the  trans- 
formers?    No  changes  are  to  be  made  in  the  windings.     (3  min.) 

34.  It  is  required  to  design  two  transformers  which  can  be  used 
on  a  6600-volt,  2-phase  system  to  give  either  2-phase  or  3-phase, 
3-wire  systems  at  220  volts;  2  volts  per  turn  are  to  be  allowed; 
the  transformers  have  2  coils  in  the  secondaries  and  a  tap  is  to  be 
used.     State  numbers  of  turns  in  coils,  location  of  tap  and  methods 
of  connecting  transformers  for  both  systems.     (4  min.) 

35.  Three  1000-  to  100-volt  transformers,  A,  B  and  C,  are  con- 
nected delta,  both  primary  and  secondary.     Taps  are  taken  out 
of  the  secondary  as  follows:  (1)  Between  A  and  B ;  (2)  86.6%  along 
B  (from  A) ;  (3)  middle  of  C;  (4)  86.6%  along  A  (from  B).     Show 
graphically  what  the  e.m.f. 's  between  (1)  and  (3)  and  between  (2) 
and  (4)  will  be  and  what  will  be  their  phase  relation.     (6  min.) 

36.  Given  two  suitably  wound  transformers  properly  connected 
to  give  400-volt,  3-phase  e.m.f.  by  the  Scott  method.     Construct 
the  T-shaped  diagram  by  the  revolution  of  which  around  the 
junction  of  the  two  lines  the  3-phase  e.m.f.'s  set  up  can  be  pro- 
duced.    By  the  rotation  of  this  diagram  construct  first  the  three 
sine  e.m.f.  curves  produced  by  the  three  vectors  making  up  the 
"T,"  then  by  the  combination  of  these  with  proper  signs  draw 
three  120-degree  e.m.f. 's  which  they  produce.     (30  min.) 

37.  The  same  as  problem  36,  except  that  the  120-degree  e.m.f.'s 


TRANSFORMERS,  GENERAL 


83 


are  to  be  plotted  directly  from  inspection  of  the  projections  of  the 
"T"  vectors.     (12  min.) 

38.  Two  1100-  to  110-volt  transformers  of  the  usual  type  and 
operated  on  a  1100-volt,  2-phase  line  can  have  the  secondaries 
connected  in  two  different  ways  to  produce  unbalanced  3-phase 
systems  in  which  two  of  the  e.m.f.'s  are  equal.     By  revolution  of 
the  appropriate  diagrams,  construct  the  e.m.f.  curves  for  these 
two  cases.     (30  min.) 

39.  The  hysteresis  curve  of  a  sample  of  steel  containing  0.184% 
silicon  is  given  by  the  following  points: 


H. 

B. 

H. 

B. 

H. 

B. 

1.4 

0 

8 

10,700 

2 

8,000 

2 

2,850 

10 

11,500 

1 

6,700 

3 

5,300 

8 

11,200 

0 

4,800 

4 

7,000 

6 

10,700 

-1 

2,100 

6 

9,400 

4 

9,600 

-1.4 

0 

If  a  transformer  is  built  of  this  steel  having  a  maximum  density 
of  11,500,  a  length  of  magnetic  circuit  of  50  centimeters  and  100 
primary  turns;  also  if  the  impressed  e.m.f.  is  a  sine  curve;  plot  the 
curves  of  primary  current  for  no  load  and  for  a  non-inductive  load 
of  10  amperes.  (30  min.) 

40.  On  the  no-load  test  of  a  transformer;  the  voltmeter  is  con- 
nected inside  of  the  ammeter  and  takes  0.05  ampere.     The  test 
shows  20  watts  iron  and  instrument  loss  and  the  e.m.f.  is  100;  find 
the  magnetizing  current  if  the  current  read  on  the  ammeter  is 
3  amperes.     (5  min.) 

41.  Construct  the  sine  waves  for  the  component  3-phase  parts 
of  the  2-phase  e.m.f.'s  obtained  by  the  connection  described  in 
problem  35  and  by  combining  these  derive  the  2-phase  sine  curves. 
The  3-phase  e.m.f.'s  are  each  100  volts.     (30  min.) 


CHAPTER  X 
TRANSFORMER  DIAGRAMS  AND   REGULATION 

Note.  —  Scale  all  quantities  asked  for.  If  problem  V-22  has  not 
been  taken,  include  here.  " Magnetizing  current"  is  taken  as  the 
magnetizing  or  wattless  component  of  the  exciting  or  no-load 
current.  The  angle  as  well  as  the  length  of  vector  is  required. 
In  all  problems  unless  otherwise  stated  or  implied  the  mean  length 
of  turn  and  the  current  density  in  the  two  windings  will  be  taken 
as  the  same.  Note  also  that  to  adapt  the  problems  to  easy 
graphical  solution  the  impedance  drop,  magnetizing  current  and 
iron  losses  have  been  exaggerated  as  compared  with  those  usually 
found  in  commercial  transformers. 

1.  A  transformer  has  600  primary  and  60  secondary  turns; 
120  magnetizing  ampere-turns  are  necessary;    draw  the  ampere- 
turn  diagrams  for  currents  of  0,  10  and  20  amperes,  secondary 
non-inductive  load,  and  find  the  primary  currents  and  lags.     (See 
note  above.)     (7  min.) 

2.  Draw  the  same  loads  both  lagging  and  leading  the  e.m.f.  by 
30  degrees,  and  find  the  primary  currents  and  lags.     (10  min.) 

3.  A  5  to  1  transformer  with  80  secondary  turns,  takes  120 
magnetizing  ampere-turns;    what  are  the  primary  currents  and 
power  factors  for  0,  4  and  8  amperes,  non-inductive  load?     (8  min.) 

4.  A  5  to  1  transformer  has  a  secondary  e.m.f.  of  2200,  a  mag- 
netizing current  in  phase  with  the  flux  of  2,  and  losses  of  5  kilo- 
watts;  draw  diagrams  for  loads  of  60  amperes,  both  when  lagging 
and  when  leading  30  degrees,  and  find  the  primary  e.m.f.,  currents 
and  power  factors.     (See  note  above.)     (13  min.) 

5.  Same  as  problem  4,  except  that  the  coils  have  resistances  of 
52  and  1.8.     Find  also  the  full-load  transformation  ratio.     Refer 
the  phase  angles  to  the  induced  e.m.f.     (25  min.) 

6.  A  10  to  1   transformer  with  220-volts  secondary  induced 
e.m.f.  and  a  load  of  400  amperes  has  1.1  kw.  iron  losses  and  a 
magnetizing  current  of  2  amperes;    draw  and  scale  diagrams  for 
secondary  power  factors  of  80%  and  40%,  and  find  the  primary 
currents.     (15  min.) 

84 


TRANSFORMER  DIAGRAMS  AND  REGULATION         85 

7.  If  in  problem  6  the  resistances  are  4.8  and  0.04,  find  the 
primary  currents  and  power  factors,  and  the  terminal  voltages. 
(30  ram.) 

8.  Draw  the  three  sine  curves  for  the  primary,  secondary  and 
magnetizing  ampere-turns  of  problem  1,  with  20  amperes  in  the 
secondary.     (25  min.) 

9.  Repeat  problem  8,  with  the  lagging  current  of  problem  2. 
(25  min.) 

10.  A  5  to  1  transformer  has  a  primary  induced  e.m.f.  of  1200 
volts,  the  resistances  are  3.1  and  0.13  ohms,  the  magnetizing  current 
3.4  amperes  and  the  iron  losses  660  watts;  draw  the  diagrams  for 
a  non-inductive  load  of  100  amperes,  and  with  the  same  load 
lagging  30  degrees  behind  the  induced  e.m.f.;    find  the  terminal 
e.m.f. 's  and  primary  currents.     (20  min.) 

11.  If  in  problem  10  the  leakage  reactances  are  6.2  and  0.26 
ohms,  draw  the  diagrams  there  called  for,  and  find  the  primary 
currents  and  the  transformation  ratios.     (30  min.) 

12.  The  secondary  current  is  10  amperes  lagging  30  degrees 
behind  the  terminal  e.m.f.     The  induced  e.m.f.  is  100  volts,  the 
secondary  resistance  1  ohm,  and  its  equivalent  reactance  2  ohms. 
Draw  the  diagram  and  determine  the  secondary  terminal  e.m.f. 
and  its  lag.     (5  min.) 

13.  A  2  to  1  transformer,  giving  500  volts  induced  secondary 
e.m.f.,  has  a  magnetizing  current  of  5  amperes,  an  iron  loss  of  1200 
watts,  secondary  resistance  and  reactance  of  5  and  10  ohms; 
construct  and  scale  the  diagram  and  determine  the  primary  current 
and  terminal  e.m.f.'s  for  a  load  having  a  resistance  of  20  and  a 
reactance  of  4  ohms.     See  note,  page  84.     (25  min.) 

14.  For  the  transformer  of  problem  13,  construct  the  sine  waves 
of  e.m.f.'s    actually  induced    by   all    flux   in   the   primary   and 
secondary  coils,  also  the  curves  of  resistance  drops  and  of  terminal 
e.m.f.     (30  min.) 

15.  A  transformer  has  150  primary  and  75  secondary  turns,  a 
magnetizing  current  of  ^  and  a  secondary  current  of  10.     The 
iron  loss  is  120  watts.     The  secondary  induced  pressure  is  100. 
Required  the  primary  current  both  when  the  secondary  circuit  is 
non-inductive  and  when  a  lag  of  30  degrees  behind  the  induced 
e.m.f.  is  present;    find  also  the  tangent  of  its  lag  in  each  case. 
(5  min.) 

16.  A  3  to  1  transformer  has  190  secondary  turns.     The  mag- 


86  ELECTRICAL  ENGINEERING  PROBLEMS 

netizing  current  is  0.8  ampere,  the  iron  loss  200  watts;  ri  is  0.9, 
r2  is  0.1;  with  300  volts  induced  e.m.f.  on  the  primary  and  20 
amperes  in  the  secondary  find  graphically  the  impressed  primary 
e.m.f.  and  the  primary  current  with  its  lag;  and  also  the  secondary 
induced  and  terminal  e.m.f. 's  for  the  following  cases:  (a)  With  the 
current  lagging  15  degrees  behind  the  induced  e.m.f.,  (b)  in  phase 
with  it,  and  (c)  leading  it  by  15  degrees.  (26  min.) 

17.  A  transformer  has  200  secondary  and  400  primary  turns; 
iron  losses,  250  watts;    ri,   1.2  and  r2,  0.35  ohm;    magnetizing 
current,  0.6;    primary  reactance,  2.4  ohms  and  secondary,  0.6. 
With  500  volts  induced  e.m.f.  in  the  primary,  construct  diagrams 
for  a  secondary  current  of  20   amperes;    (a)  in  phase  with  the 
secondary  terminal  e.m.f.,  (b)  lagging  30  degrees;  (c)  leading  30 
degrees,  relative  to  the  terminal  e.m.f.     Also  determine  the  actual 
ratio  of  transformation  in  each  case.     (50  min.) 

18.  The  secondary  induced  e.m.f.  of  a  transformer  is  200  and 
maintained  constant;  r"  equals  0.2,  x"  equals  0.6,  with  a  constant 
current  of  200  amperes.     Determine  the  secondary  terminal  e.m.f. 
for  lags  of  90,  60,  30  and  0  degrees  behind  the  induced  e.m.f.  and 
leads  of  the  same  amounts.     (10  min.) 

19.  A  2  to  1  transformer  with  constant  primary  impressed  e.m.f. 
of  100  volts  has  a  secondary  resistance  of  0.02  ohm  and  reactance 
of  0.06  ohm.     Neglect  iron  loss  and  reluctance  and  transfer  the 
primary  impedance  drop  to  the  secondary  winding;  then  determine 
the  regulations  for  a  load  of  100  amperes  with  lags  of  0,  30,  60 
and  75  degrees  behind  the  revolved  primary  terminal  e.m.f. ;   also 
for  30-  and  60-degree  lead,  and  for  unity  power  factor  of  the 
outside  circuit.     Lines  may  be  used  to  represent  the  primary  and 
secondary  impedance  drops.     (30  min.) 

20.  The  same  as  problem  19,  except  that  the  resistance  is  0.06 
ohm  and  the  reactance  0.02  ohm.     Compare  the  results  obtained 
with  those  of  problem  19.     (30  min.) 

21.  Given  a  transformer  designed  to  transform  from  300  to 
100  volts,  r"  =  0.1,  x"  =  0.2,  no-load  current  neglected.     Draw 
diagrams  for  the  following  cases:    (a)  No  load,  and  (b)  a  load  of 
50  amperes  lagging  30  degrees  behind  the  induced  e.m.f.     Under 
(b)  make  both  regular  diagrams  and  revolved  diagrams  with  all 
resistance  and  reactance  referred  to  the  primary.     Assume  (1) 
primary  terminal  e.m.f.  raised  to  keep  induced  e.m.f. 's  same  as 
at  no  load;   (2)  ditto  to  keep  secondary  terminal  e.m.f.  same  as  at 


TRANSFORMER  DIAGRAMS  AND  REGULATION        87 

no  load;   (3)  to  keep  primary  terminal  e.m.f.  unchanged.     Scale 
all  vectors  and  determine  a  regulation  for  each  case.     (20  raw.) 

22.  Draw  a  typical  diagram  for  a  constant-potential  to  constant- 
current  transformer,   (a)  with  a  large  number  of  lamps  on  the 
circuit  and  (b)  with  one  or  two  lamps.     (10  ram.) 

23.  Draw  complete  diagrams  for  a  transformer  without  iron 
core,  (a)  with  the  coils  wound  together,  turn  for  turn,  and  (b)  for 
the  coils  slightly  separated.     Explain  the  effects  of  these  condi- 
tions upon  the  form  of  the  diagram.     (15  ram.) 


CHAPTER  XI 
SYMBOLIC  EXPRESSIONS  AND   METHODS 

1.  A  current  of  10  amperes  lags  30  degrees  behind  the  impressed 
e.m.f.  of  100  volts.     Find  graphically  the  conductance,  suscept- 
ance  and  admittance  regarding  these  as  constants  which  multiplied 
by  the  e.m.f.  give  the  power  and  wattless  components  of  the  current 
and  the  current  itself.     Check  by  finding  the  relation  of  these 
quantities  to  the  resistance,  reactance  and  impedance.     (4  min.) 

2.  Required  symbolic  expressions  for  the  following  vectors  at 
the  angles  given:  (a)  3.2  cm.,  25  degrees;  (b)  6.0  cm.,  100  degrees; 
(c)  4.1  cm.,  225  degrees;  (d)  the  vector  sum  of  (a)  and  (c).     Check 
graphically.     (15  min.) 

3.  Give  symbolic  expressions  for  the  following  vectors  and  for 
their  sum:  1",  tan-1};  2",  tan"1 2;  3",  tan-!(-  1).     (5  win.) 

4.  Obtain  symbolic  expressions  for  e.m.f. 's  of  20  volts,  making 
+  30  degrees  with  horizontal;   and  30  volts  making  +  45  degrees; 
give  the  expression  for  their  sum  and  find  its  absolute  value  and 
the  tangent  of  its  angle;  also  check  graphically.     (10  min.) 

5.  The  e.m.f. 's  around  two  reactive  circuits  in  series  are  20  and 
30  volts,   and  the  power  factors  are  0.5  and  0.707.     Find  by 
symbolic  method  the  combined  e.m.f.     (3  min.) 

6.  Two  currents  of  10  and  15  amperes  in -parallel  circuits  lag 
behind  the  e.m.f.  by  angles  tan-1J  and  tan-1£ ;  by  symbolic  method 
find  sum  and  its  angle  of  lag.     (5  min.} 

7.  Two  series  circuits  having  resistances  of  3  and  4  ohms  and 
inductances  of  0.01  and  0.05  henry  are  in  series  upon  a  100-period 
system,  and  take  5  amperes;    obtain  the  symbolic  expression  for 
the  e.m.f.  and  also  its  absolute  value  and  lag.     (5  min.) 

8.  Solve  problem  5,   Chapter  V,   by  the   symbolic  method. 
(6  min.) 

9.  Solve  problem  6,   Chapter  V,   by   the   symbolic  method. 
(20  min.) 

10.  Two  parallel  circuits  have  resistances  of  8  and  3  and  re- 
actances of  6  and  4;    find  the  conductance,  susceptance  and  the 
admittance  of  the  combined  circuits;  also  with  100  volts  applied 

88 


SYMBOLIC  EXPRESSIONS  AND  METHODS  89 

the  symbolic  expression  for  the  currents  in  each  circuit  and  the 
total  current;  also  the  absolute  values  and  tangents  of  angle  of 
lag.  Check  graphically.  (10  min.) 

11.  Two  parallel  circuits  have  resistances  of  1  and  5  ohms  and 
reactances  of  7  and  5  ohms;    by  the  symbolic  method  find  the 
impressed  e.m.f.   with  a  total  current  of  100  amperes;    check 
graphically.     (6  min.) 

12.  Solve  problem  29,  Chapter  V,  by  the  symbolic  method. 
(5  min.) 

13.  Solve  problem  41,  Chapter  V,  by  the  symbolic  method. 
(15  min.) 

14.  Solve  problem  44,  Chapter  V,  by  the  symbolic  method. 
(12  min.) 

15.  Solve  problem  4,  Chapter  VI,  by  the  symbolic  method. 
(5  min.) 

16.  Solve  problem  10,  Chapter  VI,  by  the  symbolic  method. 
(8  min.) 

17.  A  500-kv-a.,  2000-volt  single-phase  transmission  line  has  a 
resistance  of  0.22  and  an  inductive  reactance  of  0.36  ohm.     What 
e.m.f.  must  be  generated  to  give  2000  volts  with  a  unity  power- 
factor  load?     (4  min.) 

18.  Repeat  problem  17,  with  the  power  factor  70.7%.     (5  min.) 

19.  The  no-load  or  exciting  current  of  a  6000  to  2200,  100-kv-a. 
transformer  is  0.47   ampere,  the  iron  loss  is  1020  watts.     By 
the  symbolic  method  determine  the  total  primary  current  for 
full-load  secondary   current  at  unity  power  factor   (referred  to 
secondary  induced  e.m.f.);  also  find  the  magnetizing  component. 
(5  min.) 

20.  Same  as  problem  16,  except  that  the  load  current  has  a 
power  factor  of  70.7%  referred  to  the  induced  e.m.f.     (4  min.) 

21.  A  20-kv-a.,  3000  to  220-volt  transformer  has  a  secondary 
resistance  of  0.0126  ohm  and  a  secondary  reactance  of  0.0110 
ohm.     With  a  secondary  induced  e.m.f.  of  220   volts   find  the 
terminal  e.m.f.  (symbolic  expression)  for  a  load  of  incandescent 
lamps,  having  a  resistance  of  2.4  ohms.     (25  min.) 

22.  Repeat  with  a  full-load  current  lagging  30  degrees  behind 
the  induced  e.m.f.     (15  min.) 

23.  The  primary  resistance  being  2.25  ohms  and  the  primary 
reactance  1.96  ohms,  find  the  primary  e.m.f.  and  hence  the  regula- 
tion in  problem  21.     Neglect  the  exciting  current.     (20  min.) 


90  ELECTRICAL  ENGINEERING  PROBLEMS 

24.  Repeat  problem  23,  with  a  power  factor  of  86.6%  referred 
to  the  induced  e.m.f .  (20  min.) 

Note.  —  Further  practice  with  the  symbolic  method  may  be 
obtained  by  solving  any  of  the  other  problems  where  graphical 
solutions  are  called  for. 


CHAPTER  XII 
ALTERNATOR  REACTIONS  AND  REGULATIONS 

1.  The  maximum  value  of  the  effective  back  ampere-turns  with 
one  ampere  through  an  alternator  armature  is  100.     The  maximum 
current  is  20;    plot  on  cross-section  paper  with  scales,  a  curve 
showing  the  effectiveness  of  the  armature  coils  in  producing  for- 
ward or  back  ampere-turns.     Show  the  poles  and  plot  the  curves 
with  position  of  the  coil,  relative  to  the  poles  as  abscissae.     Plot 
also  the  current  curve  and  by  multiplication  of  ordinates  plot  a 
curve  showing  the  instantaneous  forward  and  back  ampere-turns. 
Do  this  for:  (a)  unity  power  factor;    (b)  30  degrees  lag;  and  (c) 
30  degrees  lead  of  the  current  relative  to  the  e.m.f.     (30  min.) 

2.  A  two-phase  alternator  has  poles  occupying  half  the  pole 
pitch.     There  are  10  turns  or  20  conductors  in  each  coil  of  each 
phase;  the  maximum  current  is  100  amperes.     Lay  off  the  pole 
pieces  on  a  piece  of  cross-section  paper,  and  using  the  successive 
positions  of  the  coils  relative  to  the  pole  pieces  as  abscissae,  plot 
a  curve  showing  the  combined  number  of  forward  or  back  ampere- 
turns  due  to  the  current  in  both  phases,  and  for  each  position  of 
the  coil.     Do  this:  (a)  for  unity  power  factor;    (b)  for  30  degrees 
lagging  current;  and  (c)  for  30  degrees  leading  current.     (15  min.) 

3.  Same  as   problem  2,  except  that  the  cross  ampere-turns 
are  to  be  plotted.     (15  min.)  ^ 

4.  On  short  circuit  a  certain  50-period,  2200-volt,  440-kv-a. 
generator  requires  40-amperes  field  current  to  give  200  amperes 
its  full-load   current.     On  open  circuit  the  e.m.f.  corresponding 
to  40  amperes  excitation  is  1160  volts.     What  is  the  synchronous 
impedance?     The  resistance  being  0.5  ohm,  what  is  the  synchro- 
nous reactance?     (3  min.) 

5.  Using  the  "electro-motive  force  method  "  of  determination, 
what  would  be  the  calculated  regulation  of  the  above  machine  on 
unity  power  factor?     Also  with  30  and  60  degrees  lag?     (15  min.) 

6.  The  no-load  magnetization  curve  of  an  850-kv-a,  5000-volt, 
Y-connected  alternator  is  given  by  the  following  data: 

91 


92      ELECTRICAL  ENGINEERING  PROBLEMS 

Ampere-turns 

2000    4000    6000    8000    10,000    12,000    14,000    16,000    18,000 

Volts  (to  neutral) 

940  1900  2600  2950  3160  3340  3460  3550  3620 
3625  ampere-turns  give  full  short-circuited  current  of  98.5  amperes. 
Resistance  per  phase  (to  neutral)  0.33  ohm. 

Construct  the  above  magnetization  curve  and  the  curve  of  short- 
circuit  current  with  ampere-turns. 

Construct  also  the  pessimistic  and  optimistic  zero-power-factor 
curves  between  terminal  volts  and  field  ampere-turns  (Karapetoff, 
II,  143)  and  determine  the  two  regulations  for  5000-volts  full-load 
.zero  power  factor.  (30  min.) 

7.  Determine  also  the  pessimistic  and  optimistic  values  of 
regulation  for  unity  and  80%  power  factor.  (20  min.} 


CHAPTER  XIII 
SYNCHRONOUS  MOTORS  AND   GENERATORS 

Note.  —  All  graphical  problems  are  to  be  worked  on  large  sheets. 
Always  draw  the  current  vectors. 

1.  Two  110-volt  alternators,  for  which  co  is  400,  are  mechanically 
coupled  and  connected  in  series  upon  a  circuit  of  5  ohms  and  0.005 
henry.     How  much  power  will  be  given  by  each  machine  to  the 
circuit  when  the  e.m.f.'s  of  the  two  machines  are  30,  45  and  90 
degrees  apart?     (30  ram.) 

2.  The  same  machines  as  in  problem  1  are  used  in  the  same 
way  upon  a  circuit  of  2  ohms  resistance  and  6  ohms  reactance. 
Determine  the  power  for  the  same  phase  angles.     (30  raw.) 

3.  Two  alternators  each  generate  2000  volts;  each  has  the  arma- 
ture resistance,  2,  the  inductance,  0.07,  and  the  frequency,  60. 
They  are  run  in  parallel  and  owing  to  throttling  of  one  engine  a 
phase  displacement  (from  opposition)  of  30  degrees  results.     How 
many  kilowatts  are  being  supplied  by  the  motor  action  of  the 
synchronizing  current  and  how  much  copper  loss  results?     Solve 
graphically.     (25  raw.) 

4.  A  synchronous  motor,  running  on  a  2000-volt  line,  is  over- 
excited so  that  with  10-amperes  load  its  current  leads  the  e.m.f. 
30  degrees.     A  choke  coil  with   20-ohms  reactance   and  J-ohm 
resistance  is  placed  in  series  with  it;  what  is  the  e.m.f.  at  the  motor 
terminals?     (3  ram.) 

5.  A  synchronous  motor  is  run  on  a  440-volt,  60-period  circuit 
and  takes  20  amperes  leading  30  degrees.     If  an  induction  coil  with 
L,  0.01,  is  in  series  with  it,  what  will  then  be  the  e.m.f.  on  the 
motor?     (3  raw.) 

6.  A  synchronous  motor  running  on  a  1200-volt  system  has 
an  armature  resistance  of  1.2  and  reactance  of  5.8  ohms.     For  a 
certain  field  current,  the  motor  gives  1100  volts  counter  e.m.f. 
(neglect  the  change  in  this  due  to  armature  reactions) .     Construct 
the  parallelogram  diagrams  for  currents  of  10,  25  and  40  amperes. 
Let  the  line  e.m.f.   vector  be   common  to  all  three  diagrams. 

93 


94  ELECTRICAL  ENGINEERING  PROBLEMS 

Determine  the  power  factors  of  the  line  current  in  each  case. 
(80  ram.) 

Note  that  in  these  problems  an  open  scale  and  great  care  is 
necessary  to  get  accurate  results.  The  impedance  e.m.f.  triangle 
should  be  drawn  to  a  larger  scale  than  the  motor  e.m.f.  The 
resistances  in  these  problems  are  exaggerated  as  compared  with 
commercial  machines. 

7.  Repeat  problem  6,  with  a  higher  excitation,  giving  a  counter 
e.m.f.  of  1200  volts.     Determine  also  the  power  supplied  to  the 
motor  in  each  case.     (85  min.) 

8.  Repeat  problem  6,   with  a  counter  e.m.f.  of  1300  volts. 
Determine  also  the  power  transformed  by  the  motor  to  mechanical 
energy  in  each  case.     (35  min.) 

9.  In  the  motor  of  problem  6  with  a  load  adjusted  for  each  case 
to  40  amperes,  the  excitation  of  the  motor  is  changed  so  that  the 
e.m.f. 's  are  respectively  1050,  1150  and  1250  volts.     Construct 
the  3  diagrams  for  this  case;    determine  the  power  given  to  the 
motor,  the  copper  loss  and  the  power  transformed  in  each  case. 
(35  min.) 

10.  The  motor  of  problem  6  is  run  upon  a  transmission  line 
having  a  resistance  of  1.2  ohms,  the  generator  e.m.f.  is  maintained 
at  1200  volts;  construct  the  3  diagrams  for  a  counter  e.m.f.  of  1200 
volts  and  obtain  the  tangents  of  the  lag  or  lead  of  the  current. 
(30  min.) 

11.  In  the  motor  of  problem  6  the  resistance  is  made  2.4  and 
the  reactance  11.6,  that  is  the  impedance  is  doubled.     The  line  and 
motor  e.m.f. 's  both  being  1200  volts,  construct  the  diagrams  for 
10,  25  and  40  amperes  and  find  the  power  factors  for  each  case. 
(30  min.) 

12.  A  motor  is  excited  to  95  volts,  the  resistance  is  0.1  and  the 
reactance  0.8;   the  power  taken  by  the  motor  is  constant  and  its 
e.m.f.  is  lagging  15  degrees  behind  opposition  to  the  current;    if 
the  current  be  20  amperes,  required  the  terminal  e.m.f.     With  the 
same  terminal  e.m.f.,  what  other  motor  e.m.f.  will  give  the  same 
current?     Also,   what  motor  e.m.f.  will  make  the  power-factor 
unity?     (15  min.) 

13.  With  the  motor  of  problem  12  on  a  100-volt  circuit,  find 
the  motor  e.m.f.  to  give  minimum  current  for  a  load  at  3  kw.     With 
the  same  power  transformed  and  the  same  motor  excitation,  what 
change  in  terminal  e.m.f.  would  reduce  the  current  to  a  new  min- 


SYNCHRONOUS  MOTORS  AND  GENERATORS  95 

imum;  with  this  new  line  e.m.f.  what  further  change  in  motor 
e.m.f.  would  reduce  the  current  to  still  another  minimum;  deter- 
mine also  the  value  of  the  current  for  each  case.  (20  min.) 

14.  For  the  motor  of  problem  6,  having  a  resistance  of  1.2  and 
a  reactance  of  5.8  ohms  and  excited  to  1100  volts,  construct  the 
circle  diagrams  for  10,  25  and  40  amperes  with  1200- volts  line 
e.m.f.  and  from  these  determine  the  power  transformed  to  mechani- 
cal energy  by  the  motor  for  each  case.     Find  also  the  maximum 
load  the  motor  will  carry,  and  the  current  for  that  load.     (20  min.) 

15.  Repeat  problem  14,  with  a  motor  e.m.f.  of  1200  volts. 
(10  min.) 

16.  Repeat  problem  14,  with  the  resistance  doubled.     (10  min.) 

17.  Repeat  problem  14,  with  the  resistance  and  reactance  both 
doubled.     (10  min.) 

18.  Find  the  maximum  value  that  the  motor  e.m.f.  can  have 
with  a  line  e.m.f.  of  1200  for  each  of  the  impedances  of  problems 
14,  16  and  17.     (5  min.) 

19.  A  synchronous  motor  running  on  a  220- volt  line  has  a 
resistance  of  0.18  and  a  reactance  of  0.98,  and  is  generating  210 
volts*     Construct  the  parallelogram  diagram  for  an  input  of  4 
kw.  and  determine  the  power  factor  of  the  line  current.     With 
the  same  power  and  line  e.m.f.  find  the  motor  e.m.f. 's  to  give  unity 
and  86.6%  (leading)  power  factor.     (20  min.) 

20.  With    the    same  motor  as  in  problem   19  construct  the 
parallelogram  diagram  for  4  kw.  with  the  current  in  opposition  to 
the  motor  e.m.f.  and  determine  the  necessary  e.m.f.  of  the  motor. 
Note  that  this  gives  maximum  output  of  the  motor  for  a  given 
current  and  motor  e.m.f.     (10  min.) 

21.  For  the  motor  of  problem  14  plot  excitation  characteristics 
for  10  and  30  kw.;  use  motor  e.m.f. 's  for  abscissae  and  currents  as 
ordinates;   show  on  the  curves  the  range  within  which  the  motor 
will  run.     Use  the  circular  diagram  to  obtain  the  points.     (45 
min.) 

22.  A  motor  having  a  resistance,  0.04  ohm,  and  a  reactance, 
0.25  ohm,  is  excited  to  200  volts  and  run  on  220  volts.     Draw  the 
circle  diagrams  for  the  case  where  the  motor  is  taking  no  power  and 
where  it  breaks  from  step,  also  for  J  the  latter  load;   determine 
from  these  the  current  and  power  in  each  case;  also  what  would 
be  the  minimum  generator  e.m.f.  on  which  the  motor  could  run 
with  the  least  load;  also  with  a  generator  e.m.f.  of  220  volts  what 


96 


ELECTRICAL  ENGINEERING  PROBLEMS 


would  be  the  maximum  e.m.f.  of  the  motor  with  which  it  could 
run  as  a  motor?     (20  min.) 

23.  Two  1100-volt  alternators,  A  and  B,  each  having  a  resist- 
ance of  0.21  ohm  and  a  reactance  of  1.2  ohms  are  in  parallel  and 
are  supplying  a  circuit  having  a  resistance  of  9.79  and  a  reactance 
of  3.8  ohms.     Draw  the  parallelogram  diagram  for  the  following 
cases  obtaining  the  terminal  e.m.f.  in  each  case;    (a)  the  e.m.f. 's 
are  equal  and  opposite;  (b)  owing  to  decrease  in  the  driving  power 
machine  B  lags  2  degrees  behind  opposition  to  machine  A.     Find 
the  synchronizing  current,  the  power  it  gives  to  B,  the  total 
current  on  each  machine,  and  the  power  given  by  each.     Neglect 
the  synchronizing  component  in  getting  the  terminal  e.m.f.     Note 
that  the  currents,  while  combined  in  the  outside  circuit,  are  in 
opposition  in  the  local  circuit.     (25  min.} 

24.  A  5500-volt,   1500-kv-a.,  three-phase,  Y-connected  alter- 
nator, 83  r.p.m.  50-cycle,  is  to  be  used  as  a  synchronous  motor. 
The  resistance  is  0.3  ohm  per  phase,  that  is  to  the  neutral.     The 
reactance  is  2.7  ohms  per  phase.     Construct  the  parallelogram 
diagram,  and  find  the  motor  voltage  necessary  to  give  unity  power 
factor  on  a  5500-volt  line,  with  an  input  of  1500  kw.  into  the 
motor.     Find  also  the  current  and  power  factor  with  an  excitation 
of  5500  volts.     (30  min.) 

25.  Check  the  results  of  problem  24  by  the  symbolic  method. 
(30  min.) 

26.  Construct  the  circle  diagram  for  the  alternator  of  problem 
24  and  determine  the  maximum  load  it  will  carry  when  excited 
to  5456  volts.     (10  min.) 


CHAPTER  XIV 
SYNCHRONOUS   CONVERTERS 

1.  Draw  2  diagrams  showing  4-pole,  parallel- wound  synchro- 
nous converters  for  3  and  4  phases  respectively,  and  show  the  loca- 
tion of  the  brushes  and  connections  to  the  slip  rings  on  these 
diagrams.     (8  min.) 

2.  Construct  a  table  giving  in  per  cent  for  a  direct-current 
generator  and  for  converters  single-phase  and  3,  4  and  6-phase, 
the  relation  of  maximum  and  effective  e.m.f. 's  and  of  the  effective 
alternating  currents  in  the  armature  windings  and  in  the  leads 
from  the  brushes.     Take  the  e.m.f .  and  line  current  for  the  direct- 
current  generator  as  one  hundred  in  each  case.     (15  min.) 

3.  A  converter  gives  110  volts  direct  current;  determine  graphi- 
cally the  maximum  e.m.f.  between  adjacent  brushes  of  the  alter- 
nating-current side  if  4-phase,  also  if  3-phase  and  6-phase.     From 
these  calculate  the  effective  e.m.f.'s.     (8  min.) 

4.  In  problem  3  if  the  direct  current  is  65  amperes  and  the 
efficiency  100%,  what  is  the  effective  single-phase  line  current? 
Determine  the  effective  3,  4  and  6-phase  currents  in  the  line. 
Obtain  these  values  directly  by  use  of  the  effective  e.m.f.  's.     (8 
min.) 

5.  Converters   are  placed   upon   2000-volt   systems,  single,  3, 
4   and   6-phase.      Find   the   direct-current   e.m.f.   in  each   case. 
(3  min.) 

6.  The  converters  of  problem  5  are  fed  in  each  case  with  250 
amperes  alternating  current;  determine  the  direct  current  in  each 
case.     (15  min.) 

7.  The  converters  of  problem  5  are  each  carrying  150  amperes 
alternating   current    (sine   wave),    in   the   armature   conductors; 
required   the    alternating    line   current    and   the   direct   current 
delivered.     (9  min.) 

8.  A  double-current  generator  is  designed  to  give   230  volts 
across  the  outside  of  a  three-wire  system.     The  alternating-current 
side  supplies  3-phase  current  to  step-up  transformers.     For  what 
e.m.f.  must  they  be  designed.     If  the  machine  were  4-phase  what 

97 


98      ELECTRICAL  ENGINEERING  PROBLEMS 

would  be  the  e.m.f.  of  the  transformers  using  two  transformers; 
also  using  four  transformers?     (4  ram.) 

9.  A  3-phase  railway  converter  is  giving  57  kilowatts  at  550 
volts,  with  an  efficiency  of  93%;  what  will  be  the  alternating  cur- 
rent e.m.f.?     With  a  10  to  1  transformer  in  the  circuit  what  will 
be  the  amperes  for  each  line  of  a  3-wire  transmission?     (5  ram.) 

10.  A  single-phase  converter  gives  100  amperes  to  the  direct- 
current  circuit.     Plot  the  currents  in  three  conductors,  one  midway 
between  the  slip-ring  connections,  one  adjacent  to  one  of  these 
connections,  and  a  third  halfway  between  the  other  two.     (20  ram.) 

11.  Repeat  problem  10,  with  a  3-phase  converter.     (20  ram.) 

12.  Repeat  problem  10,  with  a  4-phase  converter.     (20  ram.) 

13.  A  single-phase  converter  gives  100  amperes  to  the  direct- 
current  circuit  with  a  lag  of  30  degrees  in  the  alternating  current. 
Plot  the  currents  in  each  of  3  conductors  of  a  segment  of  the 
armature  lying  between  two  slip  rings.     Take  the  first  conductor 
of  the  segment,  the  last  conductor  and  one  midway  between  these. 
(20  ram.) 

14.  A   3-phase   converter  gives   100   amperes   to   the   direct- 
current  circuit,  with  the  alternating  current  leading  30  degrees. 
Plot  the  current  in  each  of  3  conductors  of  a  segment  of  the 
armature  lying  between  two  slip  rings.     Take  the  first  conductor 
of  the  segment,  the  last  conductor  and  one  midway  between  these. 
(20  ram.) 

15.  A  certain  four-pole,  single-phase,  synchronous  converter  for 
220  volts  on  the  direct-current  side  has  568  conductors  and  2.63  X 
106  maxwells  per  pole.     At  what  r.p.m.  would  it  run  as  an  inverted 
converter  and  what  frequency  would  it  give  on  its  alternating- 
current  end?     When  the  machine  gives  full-load  current  with  a 
large  lag,  the  back  reaction  of  the  armature  cuts  down  the  field  flux 
to  4.1  X  105.     Neglecting  armature  drop,  at  what  speed  would  the 
converter  tend  to  run  under  these  conditions,  and  what  would  be 
the  consequences?     (6  ram.) 

16.  A    60-period,    3-phase   converter  has  in  series  with  it  a 
reactance  of  20  ohms  with  negligible  resistance;    the  line  e.m.f. 
is  1100  volts;   what  will  be  the  generated  direct-current  e.m.f.  on 
a  load  of  20  amperes  (alternating  current),  when  the  field  is  excited 
to  give  30  degrees  lag,  unity  power  factor  and  30  degrees  lead? 
(10  ram.) 

17.  A  converter  has  a  capacity  of  125  kw.  when  used  single- 


SYNCHRONOUS  CONVERTERS          99 

phase;  what  is  its  capacity  as  a  three-phase  converter;  also  as  a 
4-phase  and  a  6-phase  converter;  also  as  a  direct-current  generator? 
(10  min.) 

18.  A  certain  converter  is  wound  for  a  3-phase  system,  with 
50  pounds  of  copper;   with  the  same  heating,  what  would  be  the 
saving  in  copper,  if  this  machine  were 'to  be  connected  and  used 
for  the  same  output  on  a  4-phase  system;    also  for  a  6-phase? 
(7  min.} 

19.  Plot  as  ordinates  the  currents  for  one  period,  in  a  conductor 
TV  of  the  pole-pitch  distant  from  the  collector-ring  connection  in 
a  3-phase  converter,  giving  130  amperes  direct  current.     (5  min.) 

20.  For  an  8-phase  converter,  determine  the  e.m.f.  and  current 
relations  and  the  ratio  of  its  power  capacity  to  the  same  machine 
used  as  a  direct-current  generator.     (10  min.) 

21.  A   3-phase,   2-pole   converter  gives  250   amperes,   direct 
current ;  when  the  ring  connection  has  passed  the  brush  30  degrees, 
plot  the  curve  of  current  in  each  conductor.     (15  min.) 


CHAPTER  XV 
POLYPHASE  INDUCTION  MOTORS 

1.  Lay  off  on  two  axes,  making  60  degrees  with  each  other,  the 
successive  sets  of  field  values  due  to  a  two-phase  system  of  currents 
in  coils  at  right  angles  to  the  above  axes.     Combine  these  and 
hence  obtain  a  polar  curve  showing  the  strength  of  the  revolving 
field.     Note  also  the  angular  velocity  of  the  field  at  different 
points.     (10  mm.) 

2.  Lay  off  on  three  axes,  making  120  degrees  with  each  other, 
the  successive  sets  of  field  values  due  to  a  three-phase  system  of 
currents  in  three  coils  at  right  angles  to  these  axes.     Combine  the 
three  fields  of  each  set  and  thus  obtain  a  polar  curve  showing  the 
strength  of  the  revolving  field.     Note  also  the  angular  velocity  at 
different  points.     (15  mm.) 

3.  A  12-pole  (per  phase)  induction  motor  is  running  on  a  60- 
period  circuit;  what  is  the  frequency  of  the  current  in  the  second- 
ary, when  it  is  running  at  600,  500,  300  and  0  r.p.m.?     (4  mm.) 

4.  Construct   a  table   showing   synchronous  speeds  in  r.p.m. 
for  2-,  4-,  6-,  8-,  10-  and  12-pole  motors  at  frequencies  of  100,  60, 
40  and  25.     (9  mm.) 

5.  Draw  a  series  of  diagrams  showing  a  cross-section  perpen- 
dicular to  the  shaft,  of  the  rings  and  of  the  field  or  primary  con- 
ductors of  a  6-pole,  2-phase  motor,  and  show  by  dotted  lines  the 
positions  of  the  field  corresponding  to  6  =  0,  45,  90  and   135 
degrees  in  i0  =  Ia  sin  6.     Draw  the  coils  and  field  for  one-half  of 
the  ring  only.     Letter  the  phases  A  and  B  and  indicate  the  suc- 
cessive position  of  a  particular  pole  by  marking  with  an  X.     (15 
mm.) 

6.  Repeat  problem  5  with  a  4-pole,  3-phase  motor  and  corre- 
sponding to  0  =  0,  30,  90  and  180  degrees.     (20  mm.) 

7.  The  stator  and  rotor  of  a  440-volt  induction  motor  are  both 
Y-connected.     The  stator  current  is  18  and  the  rotor  current  108. 
If  the  slip  is  4%,  what  e.m.f.  is  set  up  in  the  rotor  circuit  between 
neutral  and  one  slip  ring?     (3  mm.) 

8.  A  300-h.p.,  440-volt,  50-cycle,  12-pole  motor  has  a  delta- 

100 


POLYPHASE  INDUCTItoN'-'MOtOSfc  101 

connected  primary  and  Y-connected  secondary.  The  full-load 
efficiency  is  92%  and  the  power  factor  is  85%.  There  are  432 
stator  and  338  rotor  conductors.  Neglecting  losses,  what  will  be 
the  full-load  currents  in  stator  and  rotor,  and  the  e.m.f.  per  phase 
(to  neutral)  in  the  rotor  when  at  standstill?  The  resistance  per 
phase  being  0.022  and  the  reactance  neglected,  what  will  be  the 
rotor  e.m.f.  and  the  slip  at  rated  load?  (10  min.) 

9.  If  the  reactance  at  standstill  of  the  rotor  of  problem  8  is 
0.20  ohm,  and  if  an  outside  resistance  of  0.218  ohm  be  connected 
by  the  slip  rings  into  each  phase,  at  what  speed  will  the  motor  run 
when  taking  full-load  current?     (9  min.) 

10.  Find  the  equivalent  single-phase  value  of  the  rotor  resist- 
ance and  of  the  rotor  and  stator  currents  at  rated  load  in  problem 
8.     (5  min.) 

11.  A    25-cycle,    200-h.p.,    1000-volt,    Y-connected   induction 
motor  has  a  wound  Y-connected  rotor  with  slip  rings  and  a  trans- 
formation ratio  of  3.6.     The  resistance  per  phase  is  0.01.     If  the 
inductance  per  phase  is  0.00064  henry,  what  would  be  the  starting 
current  with  short-circuited  slip  rings?     Also  what  would  be  its 
phase  angle  with  the  flux?     The  full-load  slip  being  3%,  what  will 
be  the  current  and  its  angle?     What  resistance  per  phase  would 
have  to  be  inserted  to  get   300-amperes  starting  current  in  the 
primary,  and  what  would  be  the  phase  angle  of  the  current  in  the 
secondary?     Neglect  the  primary  losses.     (25  min.) 

12.  The  conductors  of  an  induction  motor  are  divided  into 
groups  numbered  from  1  to  48,  all  wound  in  the  same  direction  and 
having  the  ends  marked  +  and  — .     Show  to  what  each  end  of 
each  section  must  be  connected  (that  is  to  a  main  or  the  +  or  — 
end  of  another  coil)  to  give:    (a)  a  4-pole,  2-phase  motor;    (b)  a 
12-pole,  2-phase  motor;    (c)  a  4-pole,  3-phase  motor,  delta-con- 
nected;  (d)  an  8-pole,  3-phase  motor,  Y-connected.     (25  min.) 

13.  The  total  no-load  losses  in  a  381-volt,  45-h.p.,  Y-connected 
motor  are  1416  watts.     The  magnetizing  ampere-turns  necessary 
for  the  air  gap  are  184,  and  for  the  rest  of  the  circuit  36,  maximum 
per  pole  per  phase.     The  turns  per  pole  per  phase  are  10.     What 
is  the  effective  no-load  current   and  what  is  its  power  factor? 
Assuming  the  magnetizing  current  constant  and  neglecting  leak- 
age, what  would  be  the  power  factor  with  an  input  of  20  kw.? 
Note  that,  in  this  case,  leakage  reduces  the  power  factor  about 
5%.     (25  min.) 


102  ELECTRICAL  ENGINEERING  PROBLEMS 

14.  A  440-volt  induction  motor  gives  the  following  data  on 
test:  No-load  current,  4.0;  power  factor,  0.3;  current  for  blocked 
rotor,  50;   and  power  factor,  0.5;    primary  resistance,  2.3  ohms. 
Construct  the  circle  diagram  of  the  motor  and  determine  the  rotor 
current,  power  factor,  output,  input,  efficiency,  per  cent  slip  and 
torque  in  synchronous  watts  for  a  stator  current  of  20  amperes. 
Determine  also  the  maximum  power  factor  and  torque  with  the 
corresponding  outputs.     (60  min.) 

15.  A  220-volt,  20-h.p.,  delta-connected  motor  takes  20  am- 
peres per  phase  line  current  at  no  load  and  270  amperes  with 
blocked  rotor.     The  corresponding  watts  per  phase  are  300  and 
15,000.     The   stator  resistance,  as  measured  from  line  to  line, 
is  0.1  ohm.     Construct  curves  of  line  current,  input,  efficiency, 
per  cent  slip  and  power  factor  with  horse-power  output  as  ab- 
scissae.    Get  four  sets  of  points  besides   those  for  no  load   and 
blocked   rotor.     Give   the   above   items  corresponding   to    138.5 
amperes,  line  current,  input.     (90  min.) 

16.  A  350-h.p.,  500-volt,  3-phase  motor,  300-r.p.m.,  50-cycles 
has  both  stator  and  rotor  star-connected.     It  gives  on  test  the 
following  data:  With  no  load  the  current  is  145  amperes  and  the 
power   1.1   kilowatts.     With  blocked  rotor  the  current  is  1600 
amperes  and  the  power  factor  0.28.     The  stator  resistance  per 
phase  is  0.015  ohm.     Construct  the  circle  diagram  and  by  means 
of  it  plot  the  speed-torque  curve.     Determine  the  resistance  of 
the  rotor  circuit  and  that  which  must  be  added  to  give  maximum 
torque  at  starting.     (60  min.) 

T/       r'  (r'/2  +  x2s2) 

17.  Using  the  formula  ^  =  r// (r/2  +  x2s2) ' 

where  r'  and  r"  are  two  secondary  resistances  and  x  is  the  second- 
ary reactance,  0.047,  plot  the  speed  torque  curve  for  the  motor 
of  problem  16  which  will  give  maximum  torque  at  starting.  Base 
this  upon  the  rotor  resistance  as  found  in  problem  16.  (60  min.) 


POLYPHASE  INDUCTION  MOTORS 


103 


BROWN  AND  SHARP  WIRE  TABLE 


B.  &S. 
No. 

Diameter, 

mils. 

Area, 
circular 
mils. 

B.  &S. 
No. 

Diameter, 
mils. 

Area, 
circular 
mils. 

4° 

460 

211,600 

21 

28.5 

812.3 

3° 

410 

168,100 

22 

25.3 

640.1 

2° 

365 

133,220 

23 

22.6 

510.8 

0 

325 

105,620 

24 

20.1 

404.0 

25 

17.9  ' 

320.4 

1 

289 

83,520 

26 

15.9 

252.8 

2 

258 

66,560 

27 

14.2 

201.6 

3 

229 

52,440 

28 

12.6 

158.8 

4 

204 

41,620 

29 

11.3 

127.7 

5 

182 

33,120 

30 

10.0 

100.0 

6 

162 

26,240 

31 

8.9 

79.2 

7 

144 

20,730 

32 

8.0 

64.0 

8 

128 

16,380 

33 

7.1 

50.4 

9 

114 

13,000 

34 

6.3 

39.7 

10 

102 

10,400 

35 

5.6 

31.4 

11 

90.7 

8,226 

36 

5.0 

25.0 

12 

80.8 

6,529 

37 

4.4 

19.4 

13 

72.0 

5,184 

38 

4.0 

16.0 

14 

64.1 

4,108 

39 

3.5 

12.3 

15 

57.1 

3,260 

40 

3.1 

9.6 

16 

50.8 

2,580 

17 

45.3 

2,052 

18 

40.3 

1,624 

19 

35.9 

1,288 

20 

32.0 

1,024 

Note.  —  The  system  of  sizes,  known  as  the  B.  &  S.  gauge,  represent  a  geo- 
metric series  between  No.  4/0  and  No.  36,  and  involve,  therefore,  large  num- 
bers of  decimals  for  their  exact  expressions.  In  preparing  the  above  table  the 
specifications  of  the  American  Society  for  Testing  Materials,  dated  June  1, 
1912,  were  taken  into  consideration.  These  provide  that  for  soft  wires  the 
permissible  variation  from  nominal  diameter  shall  be  for  wire  0.01  inch  in 
diameter  and  larger,  1%  over  or  under.  For  wires  less  than  0.01  inch  in 
diameter,  0.1  mil  over  or  under.  For  medium  hard  and  hard  wire  1%  varia- 
tion is  allowed  for  wires  0.1  inch  and  larger,  and  1  mil  for  smaller  wires.  In 
expressing  the  size  of  such  wires  not  more  than  three  decimals  of  an  inch  shall 
be  used,  namely,  whole  mils.  The  specifications  further  condemn  the  use  of 
large  numbers  of  decimals  in  expressing  the  diameter  of  wires,  and  recom- 
mend that  actual  diameters  rather  than  gauge  numbers  be  used. 


104 


ELECTRICAL  ENGINEERING  PROBLEMS 


MAGNET  WIRE 

For  magnet  wire  double-cotton-covered,  to  obtain  the  diameter  add  for 
wires  from 

8 


4/0  to 

9  to  1§ 

16  to  34 

35  to  37 

37  to  40 

to  the  diameter  of  the  bare  wire. 


14  mils 
11  mils 
9  mils 
8^  mils 
8  mils 


RESISTIVITY 
Mil-foot 


Temp,  in 
Cent. 

Temp,  in 
Fahr. 

Copper. 

Aluminum. 

Typical  iron. 

0 

32 

9.59 

16.0 

67.0 

5 

41 

9.78 

16.3 

69.0 

10 

50 

9.97 

16.6 

71.0 

15 

59 

10.14 

16.9 

73.0 

20 

68 

10.35 

17.3 

75.0 

25 

77 

10.55 

17.6 

77.0 

30 

86 

10.75 

17.9 

79.1 

35 

95 

10.95 

18.2 

81.1 

40 

104 

11.16 

18.5 

83.1 

45 

113 

11.36 

18.9 

85.1 

50 

122 

11.57 

19.2 

87. 

55 

131 

11.77 

19.5 

89. 

60 

140 

11.98 

19.8 

91. 

65 

149 

12.19 

20.1 

93. 

70 

158 

12.40 

20.5 

95.2 

75 

167 

12.61 

20.8 

97.2 

80 

176 

12.82 

21.1 

99.2 

85 

185 

13.03 

21.4 

101.2 

90 

194 

13.23 

21.7 

103.2 

95 

203 

13.43 

22.1 

105.2 

100 

212 

13.64 

22.4 

107.2 

POLYPHASE  INDUCTION  MOTORS 


105 


TYPICAL     B-H     OR    MAGNETOMOTIVE-FORCE     CURVES     FOR 
DIFFERENT  IRONS 


HC=M.M.F.  per  centimeter. 

Kilo- 

Kilo- 

Hc 

gausses. 

gausses. 

cast  iron. 

Sheet  steel. 

Cast  steel. 

Wrought  iron. 

3 

1.3 

2.9 

2.0 

3 

5.0 

4 

1.6 

3.4 

2.5 

3.5 

6.5 

5 

1.9 

3.9 

3.0 

4 

8.5 

6 

2.3 

4.5 

3.5 

4.5 

11.2 

7 

2.6 

5.1 

4.0 

5. 

14.5 

8 

3.0 

5.8 

4.5 

5.5 

18.5 

9 

3.5 

6.5 

5.0 

6. 

23.5 

10 

3.9 

7.5 

5.6 

6.5 

30.0 

11 

4.4 

9.0 

6.5 

7. 

38.5 

12 

5.0 

11.5 

7.9 

7.5 

49.0 

13 

6.0 

15.2 

10.0 

8. 

60 

13.5 

7.0 

18.0 

11.5 

8.5 

74 

14 

8.8  * 

21.5 

14.5 

9 

89 

14.5 

11.3 

26.0 

18.5 

9.5 

106 

15 

14.7 

32.0 

25.0 

10 

124 

15.5 

20.0 

40.0 

35.0 

10.5 

144 

16 

27.0 

49.0 

49.5 

11 

166 

16.5 

37.0 

60.0 

69.0 

11.5 

192 

17 

51.0 

73.0 

93.0 

12 

222 

17.5 

69.0 

90.0 

120 

12.5 

255 

18 

91.0 

112 

152 

13 

290 

18.5 

118 

139 

189 

13.5 

328 

19 

150 

175 

229 

14 

369 

19.5 

188 

223 

277 

20 

231 

285 

20  5 

278 

21 

329 

21.5 

383 

22 

439 

22.5 

496 

23 

553 

24 

668 

25 

783 

While  these  values  represent  fairly  the  average  results  obtained  from  good 
quality  material,  better  values  are  sometimes  obtained. 


ANSWERS  TO  ACCOMPANY 

ELECTRICAL   ENGINEERING   PROBLEMS 

F.  C.  CALDWELL 


Copyright,  1914,  by  the  McGraw-Hill  Book  Company,  Inc. 


NOTE. — These  answers  are  usually  carried  further  than  the 
data  given  would  warrant,  in  order  to  form  a  basis  of  comparison. 

PART   I 
DIRECT  CURRENT 

CHAPTER  I 

1.  35,  25,  10.  9.  0.2,   0.033,   0.1111,   0.3666,   0.5, 

2.  12,  24.  1.166;  5,  30,  9,  2.727,  0.857. 

3.  70,  240,  24.  10.  4. 

4.  0.00160,  0.00255,  0.0182,  129.55.      11.  5,  3125,  5,  29.80. 

6.  25,  1.5.  12.    41.5,  0.415. 

7.  2,  1;  2,  2000.  13.    53,  9£,  62|. 

8.  0.666,  0.0833;   1£,  12;   10,  6,  4;      14.   200. 

12,  8. 

CHAPTER  II 

1.  5.68,  22.73,  181.8,  285.7;   32.29,        7.   10.40. 

2.74%;  516.5,  1.11%;  33,060,  8.  16.00. 

0.121%,  81,630,  2.26%.  9.  1.081. 

2.  41,600,  5200,  1040.  10.  257.9. 

3.  101.5,  203.0;  13.30.  11.  139.6,  360.4. 

4.  46,875,  59,675,  59,237.  12.  0.5,  247.8,  0.0054,  zero. 
6.   4,  ]4,  30;  32,  24,  14,  7,  2.  13.  3065,  No.  15. 

6.  32.62,  25.24. 

CHAPTER  III 

1.  7.373,  11,  5.5.  6.  510.57,  2.11%,  211.4. 

2.  $17.90.  7.  80.13,  11,950,  16.03,  16.03. 

3.  6.24  cents.  8.  1.613,251.7,14.43. 

4.  42.88.  9.  80. 
6.  516.76,  3.35%,  335.2. 

CHAPTER  IV 

1.  0.02771,  153.6.  4.   0.24975,  249.75. 

2.  485,  73.49,  32,000.  6.   48,000. 

3.  0.00025,  0.25,  500. 


2     ANSWERS  TO  ELECTRICAL  ENGINEERING  PROBLEMS 


CHAPTER 

V 

2.    1500,  238.7. 

14. 

785. 

3.   756.0. 

15. 

0.1070. 

4.   3930,  2812. 

16. 

24.36. 

5.   0.007854,  600. 

18. 

5.6,  657. 

6.   0.02356,  6750. 

19. 

5.35. 

7.   940. 

20. 

38.53,  81,180. 

8.   496,  16,  160. 

21. 

14.66,  36,850. 

9.   7680. 

22. 

18.30. 

10.  0.008398,  13.55. 

23. 

1.3. 

11.   2.487,  7.025,  17.56,  36.29. 

24. 

291.6. 

12.   22.96,  183.0,  1145,  4886. 

25. 

1.278. 

13.   0.03466  cm.,  400. 

26. 

12.29  X  24.58. 

CHAPTER 

VI 

1.   1440,  No.  18. 

10. 

300. 

2.   960,  No,  20,  20,833. 

11. 

2.935",    374.3,    No.    24,    8092, 

3.   10.54. 

0.4040. 

4.   2110,  No.  16. 

12. 

1950,  1273;  5.87",  632.5,  No.  22, 

5.    1364,  No.  18. 

5097. 

6.   27.93. 

13. 

9700  per  pole,    16,130,    No.  8, 

7.    184.4,  20.96%. 

82.5. 

8.    10,700,  930,  No.  20,  1,024 

,  8320.      14. 

31.92  for  7000,  same. 

9.   4596,  1045,  No.  19,  4250. 

15. 

1.25,  7.430. 

,                                                   CHAPTER 

VII 

1.   3.24. 

256  bars,  5X5  mm.;    208.3 

2.    12.15. 

amp.,  25  kw. 

3.  0.588. 

14. 

227,272. 

4.   79.6. 

15. 

500  volts,  80  amp.,  40  kw.,  0.74%. 

5.   121.6. 

16. 

750  volts,   200  amp.,   150  kw., 

6.    1667. 

1.512%. 

7.    14.07  X  106,  253.2  sq.  in. 

17. 

255.4,  2.16%,  2.16%. 

8.   512,  2.4%. 

18. 

220  volts,  136.4  amp.,  30  kw. 

9.   33|. 

19. 

576  turns,  No.  14. 

10.   148. 

20. 

480  volts,  12.02  and  7.395  kw. 

11.   86.4. 

21. 

8.82". 

12.   94.25. 

22. 

26.2%,  9.8%. 

13.   60  volts,   416.7  amp.,   25  kw.;      23. 

122,  12.69. 

CHAPTER 

IX 

1.   100. 

6. 

0.169,  0.676,  same. 

2.   4.648. 

7. 

17.25,  9.75,  none. 

3.  0.003083. 

8. 

27.194,    No.    11    (6799   circular 

4.  0.4917,  34.68,  17.05. 

mils),  98.81. 

5.   0.1229,    69.36;     0.4917, 

34.68; 

0.01967,  173.4. 


ANSWERS  TO  ELECTRICAL  ENGINEERING  PROBLEMS     3 

CHAPTER  X 

1.  7000,  3500.  9.  400,  133i  2127. 

2.  75.6,  50.4;  1512,  1008.  10.  8333,  1389,  5.01. 

3.  1833i  3666|;  3666|,  7333i  11.  1.25,  0.163",  2.5,  0.326". 

4.  6680,  1670;  same;  26,720,  6680.  12.  5547,  2133,  30.03,  0.2582;  6187, 
6.  0.7330,0.3665.  1493,  21.02,  0.2881;   3093, 

6.  2630.  747,  10.5,  0.1441;  0.3228. 

7.  485.1.  13.  Max.  B  7000,  8786,  14,144. 

8.  71.63,  76.40.  14.  253. 

CHAPTER  XI 

1.  771.5,  1487,  2419,  3792,  6190;  3.  27%,  3550,  20%. 

1.21,  1.83,  2.79  amp.  4.  For  200,000,  Ni  =  3804. 

2.  For  flux  =  5  X  106,  Ni  =  5140,  5.  9.85%  decrease,  19.1%  increase. 

i  =  2.85,  e  =  186;    3.06,  3.73  6.  14,110,  7.85  amp.;  478  volts,  no. 

and  4.70  amp.  7.  16.38,  19.31. 

CHAPTER  XII 

1.  |,  0.083|.  10.  I,  10,  E,  21.5;  I,  60,  E,  129;  I, 

2.  110,  0.1;  102.2,  22.  150,  E,  142.5. 

3.  505,  10;  502.5,  51.  11.  I,  5.3,  E,  142.2;  I,  55.8,  E,  125.2; 

4.  I,  60,  E,  101.  I,  75.2,  E,  83.8. 

5.  118.4,  105.  12.  I,  27,  E,  217.1;  I,  14.3,  E,  192.7; 

6.  118.26,  105.14.  I,  25.8,  E,  156.3. 

7.  Reduced  23.1%.  13.  Full-load  point,  total  character- 

8.  317.0,  433.8.  istic  I,  205.6,  E,  576.5. 

9.  I,  3.67,  E,  110.4;  I,  62.0,  E,  97.4;  14.  Full-load  point,  total  character- 

I,  51.3,  E,  46.2  (3  points.)  istic  I,  828.6,  E,  569.47. 

CHAPTER  XIII 

1.  81.62  degrees.  4.  85.7  degrees. 

2.  50.26,  1.99.  6.  25.45,  922.3. 

3.  8.944  inches.  6.  48.4,  154.1. 

-..,.-.        CHAPTER  XIV 

1.  917.9,327.06.  12.  89.1,90.5,89.5,88.0,86.1. 

2.  8695.  13.  88.8. 

3.  3007.  14.  77.2. 

4.  87.57,  86.95.  16.  75.7,  9.852. 
6.   83.75.  16.  12,820. 

6.  1042.7,  5349.  17.  90.9,  93.4. 

7.  92.43,  89.1.  18.  521.2,  1737,  26.58,  91.56%. 

8.  3.691  X  10-10.  19.  90.1. 

9.  11.373  X  10-15.  20.  13.01  h.p. 

10.  Full-load  (95.6  amp.)  loss  834.  21.  78.3. 

11.  20.45  kw.  22.  Motor,  94.3%;  generator,  94.8%. 


4     ANSWERS  TO  ELECTRICAL  ENGINEERING  PROBLEMS 


CHAPTER  XV 


1.  2  X  10s. 

2.  5.742.  15. 

3.  35.33,  33.65.  .         16. 

4.  1.775.  17. 

5.  210.0,  1543.  18. 

6.  1.693,  883.5.  19. 

7.  645,  604,  6.79%.  20. 

8.  644,    562,    14.68%;     731,    685,      21. 

6.71%.  22. 

9.  643,  499,  28.85%.  23. 

10.  1031,  1016,  1.477%.  24. 

11.  0.68%,  4.39  X  106.  25. 

12.  1.207,  98.6%,  49.4%.  26. 

13.  2.414,  515,  1.212.  27. 

14.  344   conductors   90  X  200  mils,      28. 

2-path;     or    688    conductors      29. 

90  X  100  mils,  4-path;  wind- 


ings  of  2  poles  in  series. 
777. 

723,  875. 
669  r.p.m. 
2.82,  22.4  ohms. 
739  r.p.m. 
342.2,  336. 
2.96,  9.38  Ibs. 
4503. 
600;  10. 
800;  10. 
533.5;  10. 
882,  488,  418. 
272,  696. 
864. 
15,000,  815,  473. 


PART   II 


ALTERNATING   CURRENT 


CHAPTER   I 


1. 
2. 
3. 
4. 
5. 

Current. 


200  amperes  per  second. 

1.2  X  104  maxwells  per  ampere. 

7,  5,  6  miles  per  hour. 

$33-£,  -$18.50  per  rnile. 


Kilogausses  per 
ampere. 

0.25  48.2 

1  15.4 

3  5.70 

5  3.62 

6.  28,160,  17,000. 

7.  Ivolt. 

8.  10,  0.04. 

9.  3200. 

10.  34,940. 

11.  482,  616,  684,  724. 

12.  28.16. 


Kilomax- 
wells  per  amp. 

964 
308 
114 
72.4 


13.  750,  0.05. 

14.  37.5,  3750. 

15.  96,  192. 

16.  0.16. 

17.  223.9. 

18.  3686,  184.3. 

19.  0.06801. 

20.  0.06167,  0.07578. 

21.  2130,  0.1936,  3.390. 

22.  2.13,  0.616. 

23.  7.95,  6.24,  4.06,  2.57. 

24.  0.111  millihenry,  0.000296. 

25.  66,0.66. 

26.  0.8555,  1.31 6  sec. 

28.  628.6. 

29.  7.2,  0.006. 


ANSWERS  TO  ELECTRICAL  ENGINEERING  PROBLEMS     5 


CHAPTER  II 


1.  9000,  2.5  X  10s;  0.1  second. 

2.  36,000,  14,400. 

3.  1,  1.389;   2,  5.556. 
5  X  106,  50,000. 
0.025,  0.005  second 
500. 

0.75  farad,  $675,000. 

20,000. 

0.026. 

0.5. 

90,  10. 

25,  6. 


4. 

5. 

6. 

7. 

8. 

9. 
10. 
11. 
12. 


14.  952.4,  47.6. 

15.  1637      volts  at  0.001  second. 

898.5  volts  at  0.001  second. 

270.6  volts  at  0.010  second. 

16.  12.38. 

17.  0.01815  coulomb  at  0.001  sec- 

ond. 

0 . 05507  coulomb  at  0 . 004  second. 
0 .  C8647  coulomb  at  0 . 0 1 0  sr  cond. 

18.  2.545. 

19.  146. 


1.  60,  42. 

2.  60,  300. 

3.  3600,  1500. 

4.  7200,  3000. 

5.  2-pole,  3600,  1500;   p  -  20,  360 

150;  p  =  36,  200,  83 1 

6.  86.6,  70.7,  50. 


CHAPTER  III 
7.   p 


12,  f  =  65  is  the  nearest. 
Change  engine  speed  for  f  = 
60. 

9.   63.66. 
10.   70.71. 

12.  53.05,  1.061;    1;    for  sine  wave 

1.112: 1. 

13.  72.11,  1.07:1. 


CHAPTER  IV 


1.  78.02,  7.802. 

2.  11.85,    75°  58';     50;     i  =  16.76 

sin  (wt  -  75°  58'). 
78,  9.798,  10. 

0.01103  henry;   3.464,  4  ohms. 
0.02653,  0.01327. 
0.1326,  89°  56';   100;  i  =  0.1875 

sin  (a<t  —  89°  56'). 
8.66,  0.01443,  5.773. 
2309. 
9.   0.02667,  0.02. 
10.   0.03183. 

72.13,  0.0998,  61  degrees. 
144.5,  0.2003. 


11. 
12. 

13.  80,  0.01684. 

14.  36.06. 


15.  36°  52',  499.5. 

16.  Impressed,  22.36  (max.) 

17.  Impressed,  1G4.4  (max.) 

18.  Impressed.  82.5  (max.) 

19.  Impressed,  89.44  (max.) 

20.  45.36. 

21.  7.15%. 

22.  130  turns. 

23.  62.45. 

24.  0.001579,    13.16,    78°  36'; 

6.683,  84°  15'. 

25.  635.9,  852.2,  86°  21'. 

28.  2.07,  20.73;  1  :  4. 

29.  414.8,  2.653. 

30.  17.28,  1728,  5786. 

31.  E.m.f.  200  volts. 


66.7; 


1.  6013,  2088,  7024;  47.75. 

2.  85.4,  85.45,  3.85%,  52%. 

5.    15,400;    3905,    50°  11';     11,660, 
30°  58'. 


CHAPTER  V 

6.    230.1,  136.3,  261.8. 


320.5,  1269.2. 
29.13,  11.18. 
2.29. 


6     ANSWERS  TO  ELECTRICAL  ENGINEERING  PROBLEMS 


10.  16.44,  16.67. 

11.  89.14,  75.9%. 

12.  75°  9',  56°  40',  0°;  67°  18',  38.6%. 

13.  0.2755,  60  degrees,  50%,  100%. 

14.  1140.4. 

15.  98.0,  107.1;   45.45%,  22.75%. 

16.  850,  18.6%. 

17.  95.4. 

18.  84.85,  84.80,  0.0749. 

19.  90.75,  42%,  95.24%. 

20.  94.09,  87.0%. 

21.  1044,  44. 

22.  18.03,  183.6. 

23.  179.5,  26.93. 

24.  10,700,  6.55%. 

25.  230,  4360,  6375. 

26.  1232,  82.5%,  1140. 

27.  59.35,  tan-1  0.298. 

28.  139.6. 

29.  34.15,  at  68.4%. 

30.  405.7,  49°  24'. 

31.  22.45,  25°  22',  90.3%. 


32. 
33. 
34. 
35. 
36. 
37. 
39. 
40. 

41. 
42. 
43. 
44. 
45. 
46. 
47. 
48. 

49. 
50. 
51. 
52. 


CHAPTER 

1. 

1.94,  97%. 

16. 

2. 

79.0,  15.8%. 

17. 

3. 

5.66,  70.7%. 

18. 

4. 

30.47,  23°  57'  lead. 

19. 

5. 

223.6. 

20. 

6. 

147.5. 

21. 

7. 

200,  37.7,  9.8. 

22. 

8. 

20.95  sin  («  -  65°  14') 

23. 

9. 

309. 

24. 

10. 

325,  170. 

11. 

201,  12.8,  1130. 

25. 

12. 

26.6,  28.4. 

27. 

13. 

5.004,  40,  20,000,  19,975. 

15. 

146.5. 

1.  40,  10.39,  63.77. 

2.  161.7,  15.5%. 

3.  108,  84,  103.9,  60. 

4.  72.7%,  43°  20',  165. 

5.  21.65. 

6.  Lag  43°  57'. 

7.  500,  1,  9.95. 


181.7. 

8.66. 

9.39,  37°  15'. 

114.5,  151.8. 

89.4  and  21.7,  71.8%. 

13.64,  21.95. 

24.19,  97.9%. 

Transformer  104.4  and  14.90. 

Coil  105. 4  and  20. 46. 

15.5,  16.05,  43.0,  84.2%,  100.2. 

1579,  70.63,  31.58. 

101.5,  253.8,  287.1,  91.4%. 

257.5. 

453,  90.3%. 

173,  151,  77.9%. 

322.5,  96.4%. 

10,   10.5;    21.15,   37°  26';    7.51, 

0.0102. 

45.2,  67.6;  0.6,  3.48,  28°  17'. 
101.2,  25.06. 
3.38,  61.6%. 
0.345,  60%. 

VI 

38.9,  53°  15'  lead,  59.8%. 

.97.4,  15.7. 

73.2. 

614 

123,  5.24. 

42. 

21,  84. 

200,000. 

6.4,  11.93,  8.15,  47.5,  110.5, 
34°  30'  lead. 

4.995,  2,  20,000. 

0.01874;  L  =  0,  24.05  leading 
66°  56';  L  =  0.04,  19.47  lag- 
ging 69°  26'. 


CHAPTER  VII 

9.  10.5,  1.125  watts. 

10.  3.535,  2.5  and  0  kw. 

11.  8660  kw.,  $107,200. 

12.  $18,750. 

13.  05°  17'. 

14.  4.77  millihenries. 

15.  18.25  and  7.66  millihenries. 


8.   7.14  kw.;  7.14  units  above. 


f 'ANSWERS  TO  ELECTRICAL  ENGINEERING  PROBLEMS     7 


CHAPTER  VIII 


1.  42.42.  22. 

2.  176.8,  707.  23. 

3.  2309.  24. 

4.  190.5.  25. 
6.   86.6. 

6.  173.2.  26. 

7.  140.4.  27. 

8.  148,156,165;   120°  20',  114°  20', 

125°  20'.  28. 

9.  282.8,  1414.  29. 
10.    17.7.  30. 

31. 
32. 

13.  23.57,   16.67,   19.61;    13.1,  42.1,  33. 

40.0. 

14.  46.63,  45.0,  50.78;    1.30,   1.732,  34. 

1.592.  35. 

15.  8.94  amp.,  44.7%.  36. 

16.  13.8,     11.6,     10.0;     68.9,    93.1,  37. 

100.2.  38. 

17.  16.42,  9.67,  9.43;  69.6,  58.0,  48.1;  39. 

2.95,  0.91,  6.0.  40. 

18.  17.32,  10.83,  8.66,  7.85.  41. 

19.  13.6,  9.62,  11.3,  6.33.  42. 

20.  60.4,  82.7;  0.68,  1.215.  43. 

21.  344.8,  0,  216.5. 


7.85,  13.6;  23.55. 
7.76,  13.45;  41.8. 
12.15,  21.1;  65.4. 
176.8,  216.5;  200,  400;  2-phase 

200,  283;    3-phase  154,  267. 
2120,  4245,  5510. 
$167.88;      $158.40;      $167.48, 

$150.30. 
15.83,  9.14. 
50.99,  72.11. 
0.662,  1.12. 
17.32  kw.,  same. 

19.05,  18.03  kw. 

6.68,    8.85,    8.77;     4440;     79.5, 

77.8,  64.2%. 
7.57,  2165. 
Unity. 
50%. 
24%. 

53.6,  93.0. 
72.6,  74.0. 
814,  3630. 

250,  500;  115.5. 
50,  57,  73. 
115.5,  same. 


CHAPTER  IX 


1.  1  ohm;  1100,  2. 

2.  550. 

3.  0.3925,  0.09975;  8400,  12.5,  39.9. 

4.  9.89,  561. 

5.  4.03,  11;  0.0403,  1.1. 

6.  1100;  220,  110. 

7.  Series    13.65,    2.73   amp.,    1100, 

220  volts,  3.518,  0.1407  ohm, 
9.6,  1.92  volts,  1.745%;  pard- 
lel  5.46,  27.3  amp.,  550,  110 
volts,  .880,  .0352  ohm,  4.8,  .96 
volts,  1.745%. 

8.  7800,  7222,  5200,  7800. 

9.  1  to  1.914;  4  to  9. 

10.  10.4%  decrease,  16.7%;    6.65%, 

8.8%. 

11.  14.1%,  17.4%  decrease;    2.07% 

decrease. 


12.  295.4,    47.8,    343.2;     137,    19.4 

156.4. 

13.  87.5,  5.55,  93.05. 

14.  57.1,3.37,60.47;  123,8.30,131.3. 

15.  81. 

16.  98%,  97%. 

17.  95.4%,  94.3%. 

18.  0.0025;   98.2,  98.4,  98.3,  97.6%. 

19.  95.6%. 

20.  96.6,  96.8,  96.55,  94.2%;  91.4%. 

21.  329,  17.9,  346.9. 

22.  45,450;    45,450,  17,050;    45,450, 

154,550;    1  :  0.545  :  1.549. 

23.  45,450,  90,900;    45,450,  62,500; 

45,450,  200,000;  606  :  526  : 
741;  3:4;  1  :  2.13;  1  :  1.055 
in  favor  of  single  circuit  in 
each  case. 


8     ANSWERS  TO  ELECTRICAL  ENGINEERING  PROBLEMS 


24.  306.5,  102.3. 

25.  48.8. 

26.  65.5. 

27.  1.128. 

28.  127., 

29.  46.2. 

30.  75,  130;  220,  246,  246. 

31.  346.4. 

32.  57.73,  100. 


33.  220,  5  to  1. 

34.  55  turns,  tap  at  40.4  turns. 

35.  86.6  volts,  90  degrees. 


38.  110,  110,  190.5;    110,  123,  123. 

39.  Max.  I,  3.98  at  no  load,  14.75  at 

10  amp.  load. 

40.  2.993. 


10. 


CHAPTER  X 

1.019,    11°  19';    2.01, 


0.2,   90°; 

5°  45'. 
1.113,    38°  57';     2.107,    34°  43'; 

0.917,  19°  10';   1.907',  24°  30'. 
0.3,    0;     0.853,    93.9%,;     1.623, 

98.2%. 
11,000,    13.54,    93.8%;     11,000, 

11.65,  80.5%. 
11,590,     2104,     13.54,     36°  24'; 

11,590,    2104,    11.65,    20°  15'; 

5.5. 

41.6,  42.0. 

41.7,  77.7%,  2356,  214.4;    42.1, 
39.5%,  2288,  207.2. 


227,    1259,    20.8;     228.4,    1250, 
22.3. 


11.  20.8,  5.6  to  1;  22.4,  6.22  to  1. 

12.  80.8,  6°  51'. 

13.  12.77,  1555,  355.9. 

15.  5.62,  0.092;  5.8,  0.612. 

16.  7.5,    305.5,    19°  23',    100,    98.1; 

7.37,  306,  6°  17',  100,98;  7.15, 
306,  7°  10 V  100,  98.1.. 

17.  2.22  :  1;  2.12  :  1;  2.02  :  1. 

18.  88.8,    79.8,    134.2,    200;     322.6, 

299,  258. 

19.  5.26%,   20.9%,   36.6%,   34.4%; 

7.4%,  15.55%;  12.35%. 

20.  30.85%,  32.8%,  21.13%,  12.94%; 

17.2%,  2.04%;  32.08%. 

21.  19.5%,  18.7%,  23.5%. 


4. 
5. 
6. 
7. 
8. 

9. 
10. 


0.0866,  0.05,  0.1.  . 

2.9  +  j  1.35,  -1.044  -f  j  5.9088, 

-2.8987  -  j  2.8987,  0.0013  - 

j  1.54. 
0.8944    +   j    0.4472,   0.8944    + 

j  1.7888,  -2.1213  -f  j  2.1213, 

-0.3325  +  j  4.3573. 
38.53  +  j  31.213,  49.57,  0.811. 
49.5. 

24.8,  17°  24'. 

35  +  j  188.5,  191.7,  79°  28'. 
39.05,    50°  12';     116.6,    30°  58'; 

154.02,  35°  45'. 
230.2,  136.3,  261.8. 
0.2,  0.22,  0.297;    10,  0.75;    20, 

1.333;  29.7,  0.909. 


CHAPTER  XI 

11.   373.13. 


12.  34.17,  68.4%. 

13.  15.5,  16.05,  43.0;   84.2%,  100.2. 

14.  257.5. 

15.  30.47,  23°  57'  lead. 

16.  325,  to  170. 

17.  2056.9. 

18.  2102.4. 

19.  16.9;  0.438. 

20.  17.1. 

21.  218.85  -  j  1.0075;    abs.    218.84. 

22.  218.51  -  j  0.246,     abs.     218.50. 

23.  -3015  -  j  12.95,    abs.    3075.1; 

1.02%. 

24.  -3079.5  -  j  3.82,    abs,    3079.5; 

1.35%. 


ANSWERS  TO  ELECTRICAL  ENGINEERING  PROBLEMS     9 


CHAPTER  XII 


4.  5.8,  5.778. 

5.  16.96%,  37.19%;  49.44%. 


6.  24.9%,  12.9%. 

7.  17.89%,  3.74;  45.4%,  21.8. 


CHAPTER  XIII 


1.  4310,  3480;    4147,  2917;    2918, 

1250. 

2.  2037,     228.8;      2334,      -269.0; 

2421,  -1200. 

3.  37.02,  1.53. 

4.  2084. 

5.  458. 

6.  10-ampere   case    impossible    for 

this  excitation;  83.4%,  94.8%. 

7.  98.7%,  11.85  kw.;   98.99%  load, 

29.69  kw.;  99.45%  lead,  47.74 
kw. 

8.  10-ampere       case       impossible: 

62.07%,   18.25  kw.;    85.14%, 
42.38  kw. 

9.  40,  65,  1.92,  38.73;    47.73,  192, 

45.81;  45.62,  1.92,  43.7. 
10.   0.3835,  0.3378,  0.2959. 


11.  98.81,  99.68,  100.0%. 

12.  94.2,  90.6,  93.4. 

13.  99.9,  29.12. 

14.  24.27,  43.58,  181.75,  245. 

15.  11.72,  28.93,  45.8;   193.8,  256. 

16.  27.06,  43.65,  136.2,  202. 

17.  8.22,  27.5,  42.92;  90.9,  123. 

18.  5926,  3138,  5926. 

19.  93%,  217.2,  228.4. 

20.  216.2. 

22.  81,  0;    1080,  150;   379,  75;    126, 

1393. 

23.  (a)    1034;     (b)    15.6,    16.9   kw., 

62.9,  37.5;   67.6,  32.84  kw. 

24.  5454;  157.8,  99.76%. 
26.  2897. 


CHAPTER  XIV 


3.  .->•>,  67.35,  38.9. 

4.  91.92,  67.42,  50.56,  33.7. 

5.  2829,  3266,  4000,  5656. 

6.  176.8,  265.2,  353.6,  530.4. 

7.  212.2,  300;  275.6,  259.8;  300, 

212.2;  318.3,  150. 

8.  140.8,  162.6..  115. 


9.  336.8,  10.51. 

15.  884,  5667. 

16.  1372,  1672,  2030. 

17.  194.11,  238.22,  282.34. 

18.  9.24  lb.,  15.6  Ib. 

20.  0.2706,0.462. 


CHAPTER  XV 


3.  0,  10,  30,  60. 

4.  For  12-pole,  250,  400,  600,  1000. 

7.  1.694. 

8.  216.7,  480;    198.8;    10.55,  5.31. 

9.  66.16%. 

10.  0.022,831.1,650.1. 

11.  1588,    84°  17';     379.4,    16°  47'; 

0.1093,  42°  35'. 


13.  15.7,  13.67%;  90.6%. 

14.  17.3,    88.2%,    5.720   kw.,    7.722 

kw.,  74%,  10.96%,  6424. 

15.  138.5,  43.87  kw,  73%,  20.8%, 

81.6%. 

16.  0.0452,  0.0388  per  phase. 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 
BERKELEY 

Return  to  desk  from  which  borrowed. 
This  book  is  DUE  on  the  last  date  stamped  below. 


RECD 


LD  21-100m-9,'481B399sl6)476 


960S 


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